QUASI-HEREDITARY SKEW GROUP ALGEBRAS

Abstract

Given an algebra and a finite group acting on it via automorphisms, a natural object of study is the associated skew group algebra. In this article, we study the relationship between quasi-hereditary structures on the original algebra and on the corresponding skew group algebra. Assuming a natural compatibility condition on the partial order, we show that the skew group algebra is quasi-hereditary if and only if the original algebra is. Moreover, we show that in this setting an exact Borel subalgebra of the original algebra which is invariant as a set under the group action gives rise to an exact Borel subalgebra of the skew group algebra and that under this construction, properties such as normality and regularity of the exact Borel subalgebra are preserved.

1 Introduction

Quasi-hereditary algebras are algebras equipped with a partial order on the isomorphism classes of simples which fulfills certain additional properties. They were first introduced by Scott in [25], and then became a central notion in the theory of highest weight categories, initiated by Cline, Parshall, and Scott in [10]. The primary motivation of [10] came from the theory of representations of semisimple algebraic groups. Among natural examples of quasi-hereditary algebras arising from this area are the Schur algebras of symmetric groups and algebras underlying blocks of Bernstein–Gelfand–Gelfand category $\mathcal {O}$ associated with a semisimple complex Lie algebra $\mathfrak {g}$ .

Many other families of quasi-hereditary algebras of significant interest come from representation theory of finite-dimensional algebras itself. Among these are all finite-dimensional algebras of global dimension less than or equal to 2 [13, Th. 2], in particular path algebras of quivers and Auslander algebras.

In analogy to Bernstein–Gelfand–Gelfand category $\mathcal {O}$ , given a quasi-hereditary algebra A, there is a set of A-modules known as the standard modules over A, which mimics the structure and properties of Verma modules over a semisimple complex Lie algebra  $\mathfrak {g}$ . Similarly to the setting of category $\mathcal {O}$ , the category $\mathcal {F}(\Delta )$ of A-modules admitting a filtration by standard modules is of particular interest. By the Dlab–Ringel reconstruction theorem [14], the algebra A together with its quasi-hereditary structure can be reconstructed from  $\mathcal {F}(\Delta )$ .

Recall that Verma modules over $\mathfrak {g}$ are defined by induction from simple finite-dimensional modules over a Borel subalgebra. In analogy, Koenig introduced in [17] the concept of an exact Borel subalgebra B of a quasi-hereditary algebra A, which is a directed subalgebra of A such that, in particular, the induction from B-modules to A-modules maps simple B-modules to standard A-modules, so that we obtain a bijection between isomorphism classes of simple B-modules and standard A-modules. Additionally, one requires the induction functor to be exact, whence the name, which enables us to transfer homological information from $\operatorname {\mathrm {mod}} B$ to $\mathcal {F}(\Delta )$ . Together, these conditions allow us to describe the structure of $\mathcal {F}(\Delta )$ using an exact Borel subalgebra.

A fundamental theorem in the study of exact Borel subalgebras of quasi-hereditary algebras is that of Koenig, Külshammer, and Ovsienko, proved in [18], which states that every quasi-hereditary algebra is Morita equivalent to a quasi-hereditary algebra with an exact Borel subalgebra.

Quasi-hereditary algebras also feature in the work of Chuang and Kessar in [6], which was later used by Chuang and Rouquier [7] in their proof of Broué’s Abelian Defect Group conjecture for symmetric groups. There, the quasi-hereditary algebras that appear are Schur algebras corresponding to blocks of the group algebra of the symmetric group. Of central interest in this setting are the RoCK blocks. These are blocks of a given weight w which are Morita equivalent to the wreath product of the principal block with the symmetric group $S_w$ . The Schur algebras corresponding to these blocks are then also Morita equivalent to the wreath product of the Schur algebra corresponding to the principal block with the symmetric group (see [8, Th. 5.1]). This enables Chuang and Rouquier to use previous results by Chuang and Tan in [9] on the wreath products of quasi-hereditary algebras with symmetric groups, something which was later again studied by Chan in [5].

Wreath products of quasi-hereditary algebras appear again in a more recent article by Evseev and Kleshchev [15], which generalizes the result of Chuang and Rouquier from the group algebra of the symmetric group to arbitrary Hecke algebras. Here, the quasi-hereditary algebras considered are zigzag algebras.

Recall that the wreath product algebra $A \wr S_{n}$ of an algebra A with a symmetric group $S_{n}$ is isomorphic to the skew group algebra $A^{\otimes n} \ast S_{n}$ . Thus, one can hope that, after additional investigation of the structure of tensor products of quasi-hereditary algebras, results on skew group algebras of quasi-hereditary algebras may be applied to wreath product algebras of quasi-hereditary algebras.

A skew group algebra $A \ast G$ is an algebra constructed from an algebra A with an action by a group G in the following way:

• • As a $\operatorname {\mathrm {k}}$ -vector space, $A*G:=A\otimes _{\operatorname {\mathrm {k}}}\operatorname {\mathrm {k}} G$ .

• • Multiplication is given by

  $$\begin{align*}(a\otimes g)\cdot (a'\otimes g'):=ag(a')\otimes gg'. \end{align*}$$

The structure of skew group algebras, including their Morita equivalence class, their Hochschild cohomology, and their Yoneda algebra, has been studied extensively (see, e.g., [11], [20], [21], [23], [26]). The preservation of various structural properties of A under the skew group construction, such as global dimension, the property of being an Auslander algebra, or the property of being Calabi–Yau, has also been investigated by many authors, including [19], [23], [27].

In this article, we examine the relationship between possible quasi-hereditary structures on A and those on $A \ast G$ . Further, we study the relationship between the exact Borel subalgebras of the two.

Assuming a natural compatibility of the group action with the partial order, we show that $\leq _A$ induces a partial order $\leq _{A*G}$ on the isomorphism classes of simple $A*G$ -modules, and we obtain the following theorem.

Theorem 4.14. The algebra $(A, \leq _A)$ is quasi-hereditary if and only if $(A*G, \leq _{A*G})$ is quasi-hereditary.

Moreover, again assuming compatibility with the G-action, we can also relate exact Borel subalgebras of A and $A*G$ .

Theorem 4.17. Let $B\subseteq A$ be a subalgebra of A such that $g(B)=B$ for every $g\in G$ . Then $(B, \leq _B)$ is an exact Borel subalgebra of $(A, \leq _A)$ if and only if $(B*G, \leq _{B*G})$ is an exact Borel subalgebra of $(A*G, \leq _{A*G})$ .

The structure of the article is as follows. Section 2 contains a brief account of skew group algebras, including a description of the simple $A*G$ -modules in terms of simple A-modules and irreducible representations of certain subgroups of G. In Section 3, we recall some of the central results about quasi-hereditary algebras and exact Borel subalgebras. Section 4 is dedicated to the synthesis of the two concepts and contains our main results. Finally, in Section 5, we describe some exact Borel subalgebras of Auslander algebras of certain Nakayama algebras, exemplifying our methods from preceding sections.

Notation

Let $\operatorname {\mathrm {k}}$ be an algebraically closed field. All algebras are assumed to be finite-dimensional $\operatorname {\mathrm {k}}$ -algebras, and all modules are assumed to be finite-dimensional as $\operatorname {\mathrm {k}}$ -vector spaces. Unless otherwise stated, all modules are assumed to be left modules. Tensor products, if not otherwise indicated, are tensor products over $\operatorname {\mathrm {k}}$ . We denote by $D:=\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {k}}}(-, \operatorname {\mathrm {k}})$ the usual $\operatorname {\mathrm {k}}$ -duality.

For a module M and an indecomposable module N over some algebra A, we write $N|M$ if and only if N is isomorphic to a direct summand of M.

We denote by $\operatorname {\mathrm {Sim}}(A)$ a set of representatives of the isomorphism classes of the simple A-modules, and for $S\in \operatorname {\mathrm {Sim}}(A)$ we write $[M:S]$ for the multiplicity of S in M. Moreover, for any module M, we pick a projective cover $P_M$ .

For any algebra A, we denote by $\operatorname {\mathrm {rad}}$ the functor

$$ \begin{align*} \operatorname{\mathrm{rad}}:\operatorname{\mathrm{mod}} A\rightarrow \operatorname{\mathrm{mod}} A \end{align*} $$

mapping a module to its radical and by $\operatorname {\mathrm {top}}$ the functor

$$ \begin{align*} \operatorname{\mathrm{top}}:\operatorname{\mathrm{mod}} A\rightarrow \operatorname{\mathrm{mod}} A \end{align*} $$

mapping a module to its top. Recall that $\operatorname {\mathrm {top}}(M)=M/\operatorname {\mathrm {rad}}(M)$ for every $M\in \operatorname {\mathrm {mod}} A$ .

2 Skew group algebras

Throughout, let A be a finite-dimensional algebra over $\operatorname {\mathrm {k}}$ and G be a finite group acting on A such that $|G|$ does not divide the characteristic of $\operatorname {\mathrm {k}}$ . In this section, we will repeat some basic definitions and results about skew group algebras. For a more detailed introduction, see, for example, [22], [23].

Definition 2.1. For an A-module M, we define $gM:=M$ as a $\operatorname {\mathrm {k}}$ -vector spaces together with the multiplication

$$ \begin{align*} a\cdot_{gM}m:=g^{-1}(a)m. \end{align*} $$

Moreover, for an A-linear map $f:M\rightarrow N$ , we define $g(f)(m):=f(m)$ .

In this way, every $g\in G$ gives rise to an autoequivalence

$$ \begin{align*} \operatorname{\mathrm{mod}} A\rightarrow \operatorname{\mathrm{mod}} A, M\mapsto gM, f\mapsto g(f) \end{align*} $$

such that the map $G\rightarrow \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {mod}} A)$ is a group homomorphism.

The module $gM$ is also sometimes denoted ${}^gM$ (see, e.g., [23, p. 235]).

However, we have chosen this notation, so that we may identify $gM$ with the set of formal products $\{gm:m\in M\}$ and then be able to write

$$ \begin{align*} a \cdot gm=g g^{-1}(a)m. \end{align*} $$

Definition 2.2 [23, p. 224].

The skew group algebra $A*G$ is defined as

$$ \begin{align*} A*G:=A\otimes \operatorname{\mathrm{k}} G \end{align*} $$

as a $\operatorname {\mathrm {k}}$ -vector space together with the multiplication

$$ \begin{align*} (a\otimes g)\cdot (a'\otimes g'):=ag(a')\otimes gg'. \end{align*} $$

Definition 2.3. Let M be an A-module. We say that M has a G-action if there are isomorphisms of A-modules

$$ \begin{align*} \operatorname{\mathrm{tr}}_g^M:gM\rightarrow M \end{align*} $$

such that

$$ \begin{align*} \operatorname{\mathrm{tr}}_g^M\circ g(\operatorname{\mathrm{tr}}_h^M)=\operatorname{\mathrm{tr}}_{gh}^M \end{align*} $$

for all $g, h\in M$ .

Remark 2.4. Let $M, N$ be two A-modules with a G-action. Then G acts on $\operatorname {\mathrm {Hom}}_A(M, N)$ via

$$ \begin{align*} g\cdot f= \operatorname{\mathrm{tr}}_g^N\circ g(f) \circ (\operatorname{\mathrm{tr}}_g^M)^{-1}. \end{align*} $$

We call a homomorphism $f\in \operatorname {\mathrm {Hom}}_A(M, N)$ compatible with the actions on M and N, if it is a fixed point of the induced action on $\operatorname {\mathrm {Hom}}_A(M, N)$ .

Moreover, note that if $M=N$ , then the action of G on $\operatorname {\mathrm {End}}_A(M)$ is an action of algebra automorphisms.

Proposition 2.5 [22, Prop. 4.8].

There is a one-to-one correspondence between modules $(M, (\operatorname {\mathrm {tr}}_g^M)_{g\in G})$ with a G-action and $A*G$ -modules given by

$$ \begin{align*} g\cdot m:=\operatorname{\mathrm{tr}}_g^M(m), \end{align*} $$

which induces an equivalence of categories between the $A*G$ -modules and the A-modules with a G-action together with the A-linear maps compatible with this action.

Remark 2.6. Note that with this identification, the G-action on $\operatorname {\mathrm {Hom}}_A(M, N)$ for two $A*G$ -modules defined in Remark 2.4 can be written as

$$ \begin{align*} g\cdot f(m)=g(f(g^{-1}m)). \end{align*} $$

Definition 2.7. Let M be an A-module. Then we define an $A*G$ -module $\operatorname {\mathrm {k}} G\otimes M$ via

$$ \begin{align*} g'\cdot (g\otimes m):=g'g\otimes m\\ a(g\otimes m):=g\otimes g^{-1}(a)m \end{align*} $$

for $g, g'\in G, m\in M, a\in A$ .

Moreover, if H is a subgroup of G and M is an $A*H$ module, then M is in particular a $\operatorname {\mathrm {k}} H$ -module, so that we can define an $A*G$ -module $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H} M$ in the same way.

Remark 2.8. Note that if H is a normal subgroup of G, then the G-action on A induces a G-action on $A*H$ via

$$ \begin{align*} g(a\otimes h)= g(a)\otimes ghg^{-1}. \end{align*} $$

Definition 2.9. Let M be an $A*G$ -module, and let V be a $\operatorname {\mathrm {k}} G$ -module. Then we define an $A*G$ -module $M\otimes V$ via

$$ \begin{align*} g\cdot (m\otimes v):=gm\otimes gv\\ a(m\otimes v):=am\otimes v \end{align*} $$

for $g\in G, m\in M, v\in V$ and $a\in A$ .

If $f:M\rightarrow N$ is a homomorphism of $A*G$ -modules, then

$$ \begin{align*}f\otimes \operatorname{\mathrm{id}}_V:M\otimes V\rightarrow N\otimes V\end{align*} $$

is a homomorphism of $A*G$ -modules, and

$$ \begin{align*} -\otimes V:\operatorname{\mathrm{mod}} A*G\rightarrow \operatorname{\mathrm{mod}} A*G, M\mapsto M\otimes V, f\mapsto f\otimes \operatorname{\mathrm{id}}_V \end{align*} $$

defines an additive functor.

Moreover, note that $- \otimes (V\oplus V')\cong -\otimes V \oplus -\otimes V'$ and $-\otimes (V\otimes V')\cong (-\otimes V)\otimes V'$ .

Definition 2.10. We denote by

$$ \begin{align*} I_G:\operatorname{\mathrm{mod}} A\rightarrow \operatorname{\mathrm{mod}} A*G, M\mapsto \operatorname{\mathrm{k}} G\otimes M, f\mapsto \operatorname{\mathrm{id}}_{\operatorname{\mathrm{k}} G}\otimes f \end{align*} $$

the induction functor along G.

We denote by

$$ \begin{align*} R_G:\operatorname{\mathrm{mod}} A*G\rightarrow \operatorname{\mathrm{mod}} A, M\mapsto _{A|}M \end{align*} $$

the canonical restriction functor. Moreover, if H is a subgroup of G, we denote by

$$ \begin{align*} I_{G/H}:\operatorname{\mathrm{mod}} A*H\rightarrow \operatorname{\mathrm{mod}} A*G, M\mapsto \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H} M, f\mapsto \operatorname{\mathrm{id}}_{\operatorname{\mathrm{k}} G}\otimes f \end{align*} $$

the induction functor from $A*H$ to $A*G$ and by

$$ \begin{align*} R_{G/H}:\operatorname{\mathrm{mod}} A*G\rightarrow \operatorname{\mathrm{mod}} A*H, M\mapsto _{A*H|}M \end{align*} $$

the canonical restriction functor.

Lemma 2.11 [23, Th. 1.1C].

We have $\operatorname {\mathrm {rad}}(A*G)=\operatorname {\mathrm {rad}}(A)\otimes \operatorname {\mathrm {k}} G$ .

The content of the following proposition is essentially a compilation of results in [23]. However, for the sake of convenience, we will give a quick proof.

Proposition 2.12. Let V be an indecomposable $\operatorname {\mathrm {k}} G$ -module. Then the following statements hold:

1. 1. For any $A*G$ -module M,

   $$ \begin{align*} I_GR_G(M)\cong M\otimes \operatorname{\mathrm{k}} G; \end{align*} $$
   in other words,
   $$ \begin{align*} \operatorname{\mathrm{k}} G\otimes M\cong M\otimes \operatorname{\mathrm{k}} G. \end{align*} $$
   More precisely, there is a natural equivalence
   $$ \begin{align*} I_GR_G\cong -\otimes \operatorname{\mathrm{k}} G. \end{align*} $$

2. 2. For any A-module M,

   $$ \begin{align*} R_GI_G(M)\cong \bigoplus_{g\in G} gM. \end{align*} $$
   More precisely, there is a natural equivalence
   $$ \begin{align*} R_GI_G\cong \bigoplus_{g\in G}g. \end{align*} $$

3. 3. $R_G, I_G$ , and $-\otimes V$ are additive.

4. 4. $R_G, I_G$ , and $-\otimes V$ are exact and reflect exact sequences.

5. 5. $R_G, I_G$ , and $-\otimes V$ preserve and reflect projective modules.

6. 6. $R_G, I_G$ , and $-\otimes V$ preserve and reflect injective modules.

7. 7. $R_G, I_G$ , and $-\otimes V$ preserve and reflect semisimple modules.

8. 8. We have natural isomorphisms $I_G\circ \operatorname {\mathrm {top}}\cong \operatorname {\mathrm {top}}\circ I_G$ , $R_G\circ \operatorname {\mathrm {top}}\cong \operatorname {\mathrm {top}}\circ R_G$ , and $(-\otimes V)\circ \operatorname {\mathrm {top}}\cong \operatorname {\mathrm {top}}\circ (-\otimes V)$ .

Proof.

1. 1. Let M be an $A*G$ -module. Then we define an isomorphism

   $$ \begin{align*} \alpha_M:I_GR_G(M)=\operatorname{\mathrm{k}} G\otimes_{A|}M\rightarrow M\otimes \operatorname{\mathrm{k}} G, g\otimes m\mapsto gm\otimes g. \end{align*} $$
   It is easy to check that this is an isomorphism of $A*G$ -modules and that $\alpha =(\alpha _M)_M$ defines a natural isomorphism.

2. 2. Let M be an A-module. Then we define an isomorphism

   $$ \begin{align*} \beta_M:R_GI_G(M)=_{A|}\operatorname{\mathrm{k}} G\otimes M\rightarrow \bigoplus_{g\in G}gM, g\otimes m\mapsto (\delta_{gg'}m)_{g'\in G}, \end{align*} $$
   where $\delta _{gg'}$ is the Kronecker delta of g and $g'$ . It is easy to check that this is an isomorphism of A-modules and that $\beta =(\beta _M)_M$ defines a natural isomorphism.

3. 3. This is obvious.

4. 4. $R_G$ is a restriction functor and thus exact. Moreover, since tensor products over $\operatorname {\mathrm {k}}$ are exact, $I_G$ and $-\otimes V$ are exact.

5. 5. Since A is an $A*G$ -module via $g\cdot a=g(a)$ ,

   $$ \begin{align*}I_G(A)\cong A\otimes \operatorname{\mathrm{k}} G=A*G.\end{align*} $$
   Since the projective modules in $\operatorname {\mathrm {mod}} A$ are exactly the modules isomorphic to direct sums of direct summands of A, and the projective modules in $\operatorname {\mathrm {mod}} A*G$ are exactly the modules isomorphic to direct sums of direct summands of $A*G$ , this implies that $I_G$ preserves projectives.

   On the other hand, if M is an A-module such that $I_G(M)$ is projective, then so is $R_GI_G(M)\cong \bigoplus _{g\in G}gM$ , and since $M=eM$ is a direct summand of $\bigoplus _{g\in G}gM$ , this implies that M is projective. Similarly,

   $$ \begin{align*}R_G(A*G)\cong \bigoplus_{g\in G}gA\cong |G|A.\end{align*} $$
   Since the projective modules in $\operatorname {\mathrm {mod}} A$ are exactly the modules isomorphic to direct sums of direct summands of A, and the projective modules in $\operatorname {\mathrm {mod}} A*G$ are exactly the modules isomorphic to direct sums of direct summands of $A*G$ , this implies that $R_G$ preserves projectives. Moreover, if M is an $A*G$ -module such that $R_GM$ is projective, then so is $I_GR_G(M)\cong M\otimes \operatorname {\mathrm {k}} G$ and since $\operatorname {\mathrm {k}} |\operatorname {\mathrm {k}} G$ ,
   $$ \begin{align*}M\cong M\otimes \operatorname{\mathrm{k}} |M\otimes \operatorname{\mathrm{k}} G.\end{align*} $$
   This implies that M is projective.

   Finally, since V is an indecomposable projective $\operatorname {\mathrm {k}} G$ -module, it is isomorphic to a direct summand of $\operatorname {\mathrm {k}} G$ , so that

   $$ \begin{align*}-\otimes V|-\otimes \operatorname{\mathrm{k}} G\cong I_GR_G,\end{align*} $$
   and thus $-\otimes V$ preserves and reflects projectives.

6. 6. Since $\operatorname {\mathrm {D}} \operatorname {\mathrm {k}} G\cong \operatorname {\mathrm {k}} G$ as a $\operatorname {\mathrm {k}} G$ -module, this is analogous to the previous statement replacing A by $\operatorname {\mathrm {D}} A$ and $A*G$ by $\operatorname {\mathrm {D}} A*G$

7. 7. Let S be a semisimple A-module. Then $\operatorname {\mathrm {rad}}(A)S=(0)$ and hence

   $$ \begin{align*} \operatorname{\mathrm{rad}}(A*G)I_G(S)=(\operatorname{\mathrm{rad}}(A)\otimes \operatorname{\mathrm{k}} G)\operatorname{\mathrm{k}} G\otimes S =\operatorname{\mathrm{rad}}(A)(\operatorname{\mathrm{k}} G\otimes S)\\ =\sum_{g\in G}\operatorname{\mathrm{k}} G\otimes (g(\operatorname{\mathrm{rad}}(A))S) =\sum_{g\in G}\operatorname{\mathrm{k}} G\otimes (\operatorname{\mathrm{rad}}(A)S) =(0), \end{align*} $$
   where the first equality follows from Lemma 2.11 and $g(\operatorname {\mathrm {rad}}(A))=\operatorname {\mathrm {rad}}(A)$ since G acts on A via algebra automorphisms. Thus, $I_G(S)$ is semisimple.

   On the other hand, if S is a simple $A*G$ module, then $\operatorname {\mathrm {rad}}(A*G)S=(0)$ and hence

   $$ \begin{align*} \operatorname{\mathrm{rad}}(A)R_G(S)=\operatorname{\mathrm{rad}}(A)S \subseteq \operatorname{\mathrm{rad}}(A*G)S =(0) \end{align*} $$
   so that $R_G(S)$ is semisimple. Thus, $R_G$ preserves and reflects semisimple modules.

   Since for every $A*G$ -module M we have

   $$ \begin{align*} R_G(M\otimes V)\cong R_G(M)\otimes V\cong \dim_{\operatorname{\mathrm{k}}}(V) R_G(M), \end{align*} $$
   this implies that $-\otimes V$ also preserves and reflects semisimple modules.

   Moreover, if S is an A-module such that $I_G(S)$ is semisimple, then so is $R_GI_G(S)$ and hence S, since S is a direct summand of $R_GI_G(S)\cong \bigoplus _{g\in G}gS$ ; and if S is an $A*G$ -module such that $R_G(S)$ is semisimple, then so is $I_GR_G(S)$ and hence S, since S is a direct summand of $I_GR_G(S)\cong S\otimes \operatorname {\mathrm {k}} G$ .

8. 8. Let M be an A-module. Then, by Lemma 2.11,

   $$ \begin{align*} \operatorname{\mathrm{rad}}(I_GM)=\operatorname{\mathrm{rad}}(A*G)I_GM=(\operatorname{\mathrm{rad}}(A)\otimes \operatorname{\mathrm{k}} G)(\operatorname{\mathrm{k}} G\otimes M). \end{align*} $$
   Since G acts on A via algebra automorphisms, we have $g(\operatorname {\mathrm {rad}}(A))=\operatorname {\mathrm {rad}}(A)$ for every $g\in G$ . Thus,
   $$ \begin{align*} (\operatorname{\mathrm{rad}}(A)\otimes \operatorname{\mathrm{k}} G)(\operatorname{\mathrm{k}} G\otimes M)=\operatorname{\mathrm{k}} G\otimes \operatorname{\mathrm{rad}}(A)M=\operatorname{\mathrm{k}} G\otimes \operatorname{\mathrm{rad}}(M)=I_G(\operatorname{\mathrm{rad}}(M)). \end{align*} $$

   Since $I_G$ is exact, we thus have $I_G\circ \operatorname {\mathrm {top}}\cong \operatorname {\mathrm {top}}\circ I_G$ .

   On the other hand, let M be an $A*G$ -module. Then

   $$ \begin{align*} R_G(\operatorname{\mathrm{rad}}(M))=R_G((\operatorname{\mathrm{rad}}(A)\otimes G)M)=R_G(\operatorname{\mathrm{rad}}(A)M)=\operatorname{\mathrm{rad}}(A)R_G(M), \end{align*} $$

   so that, since $R_G$ is exact, we have $R_G\circ \operatorname {\mathrm {top}}\cong \operatorname {\mathrm {top}}\circ R_G$ .

   Finally, let M be an $A*G$ -module. Then, by Lemma 2.11, we have

   $$ \begin{align*} \operatorname{\mathrm{rad}}(M\otimes V)=\operatorname{\mathrm{rad}}(A*G) (M\otimes V)=(\operatorname{\mathrm{rad}}(A)\otimes \operatorname{\mathrm{k}} G)(M\otimes V)\\ =(\operatorname{\mathrm{rad}}(A*G)M)\otimes V=\operatorname{\mathrm{rad}}(M)\otimes V. \end{align*} $$

   Thus, since $-\otimes V$ is exact, we have $(-\otimes V)\circ \operatorname {\mathrm {top}}\cong \operatorname {\mathrm {top}}\circ (-\otimes V)$ .

Corollary 2.13. Let H be a subgroup of G, and let $Z\subseteq G$ be a set of representatives of $G/H$ . Then the following statements hold:

1. 1. $I_{G/H}\circ I_H$ is naturally equivalent to $I_G$ .

2. 2. $R_H\circ R_{G/H}$ is naturally equivalent to $R_G$ .

3. 3. $R_{G/H}$ and $I_{G/H}$ are additive.

4. 4. $R_{G/H}$ and $I_{G/H}$ are exact and reflect exact sequences.

5. 5. $R_{G/H}$ and $I_{G/H}$ preserve and reflect projective modules.

6. 6. $R_{G/H}$ and $I_{G/H}$ preserve and reflect injective modules.

7. 7. $R_{G/H}$ and $I_{G/H}$ preserve and reflect semisimple modules.

Moreover, if H is a normal subgroup, then we additionally have

1. 8. For any $A*H$ -module M

   $$ \begin{align*} R_{G/H}I_{G/H}(M)\cong \bigoplus_{z\in Z} zM. \end{align*} $$
   More precisely, there is a natural equivalence
   $$ \begin{align*} R_{G/H}I_{G/H}\cong \bigoplus_{z\in Z}z. \end{align*} $$

Proof.

1. 1.–3. These are obvious.

2. 4. $R_{G/H}$ is a restriction functor and thus exact. Moreover, $\operatorname {\mathrm {k}} H$ is semisimple, so that tensoring over $\operatorname {\mathrm {k}} H$ is exact.

3. 5.–7. Since

   $$ \begin{align*} I_{G/H}| I_{G/H}\circ (-\otimes \operatorname{\mathrm{k}} H)\cong I_{G/H}\circ I_H\circ R_H\cong I_G\circ R_H \end{align*} $$
   and
   $$ \begin{align*} R_{G/H}| (-\otimes \operatorname{\mathrm{k}} H)\circ R_{G/H}\cong I_H\circ R_H\circ R_{G/H}\cong I_{H}\circ R_G, \end{align*} $$
   this follows from Proposition 2.12(5–7) for H and G.

4. 8. This follows analogously to 2. in Proposition 2.12.

Corollary 2.14. Let $L'$ be a simple $A*G$ -module. Then there is a simple A-module L such that $L'|I_GL$ . On the other hand, if L is a simple A-module, then there is a simple $A*G$ -module $L'$ such that $L|R_GL'$ .

Proof. Let $L'$ be a simple $A*G$ -module. Then $R_GL'\cong \bigoplus _{L\in \operatorname {\mathrm {Sim}}(A)}[R_GL':L]L$ is semisimple and $L'$ is a summand of

$$ \begin{align*} L'\otimes \operatorname{\mathrm{k}} G\cong I_GR_G L'\cong \bigoplus_{L\in \operatorname{\mathrm{Sim}}(A)}[R_GL':L]I_GL. \end{align*} $$

Since $L'$ is simple, this implies that there is some $L\in \operatorname {\mathrm {Sim}}(A)$ such that $L'|I_GL$ .

The second statement is analogous.

2.1 An explicit description of the simples

Note that Definition 2.1 tells us that G acts on $\operatorname {\mathrm {mod}} A$ via autoequivalences. In particular, $gL$ is simple for any simple A-module L, so that we obtain an induced action of G on $\operatorname {\mathrm {Sim}}(A)$ . For $L\in \operatorname {\mathrm {Sim}}(A)$ denote by $H_L$ the stabilizer of the isomorphism class of L in G and let $Z_L$ be a set of representatives of $G/H_L$ .

In this subsection, we will give an explicit description of the simples of $A*G$ in terms of simple A-modules L and simple representations of the corresponding stabilizers $H_L$ , rectifying a result in [21].

This description is not needed for our main results, but will make it possible to obtain an explicit description of the standard modules of $A*G$ (see Lemma 4.11).

Lemma 2.15. For every isomorphism class of simple A-modules, there exists a representative L which is $H_L$ -equivariant.

Proof. Clearly, we can assume that $G=H_L$ . Moreover, since $\operatorname {\mathrm {rad}}(A)$ acts as zero on L, we can assume that A is semisimple. In this case, A is a direct product of matrix rings. Again, the matrix rings not corresponding to L act via zero, so we can assume that $A=\operatorname {\mathrm {Mat}}_n(\operatorname {\mathrm {k}})$ and $L=\operatorname {\mathrm {k}}^n$ .

Now G acts on A via automorphisms, but since A is a matrix ring, all of these are inner, so we obtain a group homomorphism $\varphi :G\rightarrow \operatorname {\mathrm {Gl}}_n(\operatorname {\mathrm {k}})$ . Hence, L obtains the structure of an $A*H_L$ -module via $\operatorname {\mathrm {tr}}_g:gL\rightarrow L, x\mapsto \varphi (g)x$ .

The following proposition is a rectification of Lemma 2 in [21].

Proposition 2.16. The simple modules of $A*G$ are exactly the modules of the form

$$ \begin{align*}\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V)\end{align*} $$

for some irreducible $\operatorname {\mathrm {k}} H_L$ -module V and an $H_L$ -equivariant simple A-module L. Two modules $ \operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(L\otimes V)$ and $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_{L'}}( L'\otimes W)$ of this form are isomorphic if and only if there is $g\in G$ such that $gL\cong L'$ and $gV\cong W$ .

Proof. First, we show that a module of the form $ \operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(L\otimes V)$ is indecomposable if V is indecomposable. First, note that, using Corollary 2.13(8), we have an isomorphism

$$ \begin{align*} \operatorname{\mathrm{End}}_{A*H_L}(\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V)) \rightarrow \bigoplus_{z\in Z_L}\operatorname{\mathrm{Hom}}_{A*H_L}(z(L\otimes V), \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V)),\\ f\mapsto (f_z:z(L\otimes V)\rightarrow \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V), zx\otimes v\mapsto f(zx\otimes v))_{z\in Z} \end{align*} $$

of vector spaces. It is easy to see that this is in fact an isomorphism of $\operatorname {\mathrm {k}} G$ -modules, where the G-action on the left is given by conjugation, and the G-action on the right is given by

$$ \begin{align*} g(f_z:z(L\otimes V)\rightarrow \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V))_{z\in Z}\\ =(f^{\prime}_z:z(L\otimes V)\rightarrow \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V), z(x\otimes v)\mapsto g(f_{g^{-1}z}(g^{-1}z(x\otimes v))))_{z\in Z}. \end{align*} $$

Moreover, since $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(L\otimes V)\cong g(\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(L\otimes V))$ as an $A*H_L$ -module, we have an isomorphism

$$ \begin{align*} \bigoplus_{z\in Z_L}\operatorname{\mathrm{Hom}}_{A*H_L}(z(L\otimes V), \operatorname{\mathrm{k}} G&\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V))\\ &\rightarrow \bigoplus_{z\in Z_L}\operatorname{\mathrm{Hom}}_{A*H_L}(z(L\otimes V), z(\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V))),\\ (f_z:z(L\otimes V)\rightarrow \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(&L\otimes V))_{z\in Z}\\ &\mapsto (f^{\prime}_z:z(L\otimes V)\rightarrow z\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V), x\mapsto z(z^{-1}f_z(x))). \end{align*} $$

This is again an isomorphism of $\operatorname {\mathrm {k}} G$ -modules, where the G-action on the right is given by

$$ \begin{align*} g(f_z:z(L\otimes V)\rightarrow z\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V))_{z\in Z}\\=(gf_{g^{-1}z}:z(L\otimes V)\rightarrow z\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V))_{z\in Z}. \end{align*} $$

Furthermore, we have an isomorphism

$$ \begin{align*} \bigoplus_{z\in Z_L}\operatorname{\mathrm{Hom}}_{A*H_L}(z(L\otimes V), & z(\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V))) \\&\rightarrow \bigoplus_{z\in Z_L}z\operatorname{\mathrm{Hom}}_{A*H_L}(L\otimes V, \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V)),\\ (f_z:z(L\otimes V)\rightarrow z(\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}&(L\otimes V)))_{z\in Z}\\ &\mapsto (z(z^{-1}f_z:L\otimes V\rightarrow \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V)))_{z\in Z}, \end{align*} $$

which is again an isomorphism of $\operatorname {\mathrm {k}} G$ -modules, where the G-action on the right is given by

$$ \begin{align*} g(z(f_z:L\otimes V\rightarrow \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V)))_{z\in Z}=(z(f_{g^{-1}z}:L\otimes V\rightarrow \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V)))_{z\in Z}. \end{align*} $$

Finally, we have an isomorphism

$$ \begin{align*} \bigoplus_{z\in Z_L}z\operatorname{\mathrm{Hom}}_{A*H_L}(L\otimes V, \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V)) &\rightarrow \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}\operatorname{\mathrm{Hom}}_{A*H_L}(L\otimes V, \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}L\otimes V),\\ (z(f_z:L\otimes V\rightarrow \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V)))_{z\in Z}&\mapsto \sum_{z\in Z}z\otimes f_z \end{align*} $$

of $\operatorname {\mathrm {k}} G$ -modules, where the G-action on the right is given by left multiplication in $\operatorname {\mathrm {k}} G$ . Now if $-^G$ denotes the fix point functor under the G-action,

$$ \begin{align*} \operatorname{\mathrm{End}}_{A*G}(\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V)) &=\operatorname{\mathrm{End}}_{A*H_L}(\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V))^G\\ &\cong \left( \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}\operatorname{\mathrm{Hom}}_{A*H_L}(L\otimes V, \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V))\right)^G\\ &\cong \operatorname{\mathrm{Hom}}_{A*H_L}(L\otimes V, \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V)). \end{align*} $$

Note that for $z\notin H_L$ ,

$$ \begin{align*} \operatorname{\mathrm{Hom}}_A(L\otimes V, z(L\otimes V))\cong \dim(V)^2\operatorname{\mathrm{Hom}}_A(L, zL)=0 \end{align*} $$

so that for $z\notin H_L$ ,

$$ \begin{align*} \operatorname{\mathrm{Hom}}_{A*H_L}(L\otimes V, z(L\otimes V))\subseteq \operatorname{\mathrm{Hom}}_A(L\otimes V, z(L\otimes V))=(0). \end{align*} $$

Thus,

$$ \begin{align*} \operatorname{\mathrm{Hom}}_{A*H_L}(L\otimes V, \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V)) \cong \bigoplus_{z\in Z_L}\operatorname{\mathrm{Hom}}_{A*H_L}(L\otimes V, z(L\otimes V))\\ \cong \operatorname{\mathrm{End}}_{A*H_L}(L\otimes V) \cong \operatorname{\mathrm{End}}_A(L\otimes V)^{H_L} \cong (\operatorname{\mathrm{End}}_A(L)\otimes \operatorname{\mathrm{End}}_{\operatorname{\mathrm{k}}}(V))^{H_L}. \end{align*} $$

Since $\operatorname {\mathrm {k}}=\operatorname {\mathrm {End}}_{A*H_L}(L)=\operatorname {\mathrm {End}}_A(L)^{H_L}$ , we have $\operatorname {\mathrm {End}}_A(L)=\operatorname {\mathrm {k}} $ with the trivial G-action, so the above is isomorphic to

$$ \begin{align*} (\operatorname{\mathrm{k}} \otimes \operatorname{\mathrm{End}}_{\operatorname{\mathrm{k}}}(V))^{H_L} \cong\operatorname{\mathrm{End}}_{\operatorname{\mathrm{k}} H_L}(V) \cong \operatorname{\mathrm{k}}. \end{align*} $$

Hence, $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}L\otimes V$ is indecomposable. Since it is semisimple by Corollary 2.13(7), it is thus simple.

To see that these are up to isomorphism all simple $A*G$ -modules, note that by Corollary 2.14, every simple $A*G$ -module is a summand of $\operatorname {\mathrm {k}} G\otimes L$ for some $L\in \operatorname {\mathrm {Sim}}(A)$ and

$$ \begin{align*} \operatorname{\mathrm{k}} G \otimes L\cong \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}\operatorname{\mathrm{k}} H_L \otimes L\cong \operatorname{\mathrm{k}} G \otimes_{\operatorname{\mathrm{k}} H_L} (L\otimes \operatorname{\mathrm{k}} H_L). \end{align*} $$

So decomposing $\operatorname {\mathrm {k}} H_L$ into indecomposable summands yields the claim.

Clearly, we have isomorphisms of $A*G$ -modules

$$ \begin{align*} \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L} L\otimes V\rightarrow \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L} gL\otimes gV, h\otimes x\otimes v\mapsto hg\otimes x\otimes v. \end{align*} $$

Finally, suppose we have an isomorphism of $A*G$ -modules

$$ \begin{align*} \varphi:\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V)\rightarrow \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_{L'}}(L'\otimes W). \end{align*} $$

Then, restricting to A, we obtain an isomorphism

$$ \begin{align*} R_G(\varphi):R_G(\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V))\rightarrow R_G(\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_{L'}}(L'\otimes W)). \end{align*} $$

Since

$$ \begin{align*} &R_G(\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V))\cong \bigoplus_{z\in Z_L}\dim_{\operatorname{\mathrm{k}}}(V) zL\\ \text{ and } &R_G(\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_{L'}}(L'\otimes W))\cong \bigoplus_{z'\in Z_{L'}}\dim_{\operatorname{\mathrm{k}}}(W) zL', \end{align*} $$

the theorem of Krull–Remak–Schmidt thus yields a $g\in G$ such that $gL\cong L'$ . In particular, $H_L=H_{L'}$ and, since $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L} L\otimes g^{-1}W\cong \operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L} gL\otimes W$ , we have an isomorphism

$$ \begin{align*} \varphi':\operatorname{\mathrm{k}} G\otimes_{H_L} (L\otimes V)\rightarrow \operatorname{\mathrm{k}} G\otimes_{H_L} (L\otimes g^{-1}W). \end{align*} $$

We can restrict this to $A*H_L$ to obtain an isomorphism

$$ \begin{align*} \bigoplus_{z\in Z_L}z(L\otimes V)\rightarrow \bigoplus_{z\in Z_L}z(L\otimes g^{-1}W). \end{align*} $$

Since $L\otimes V$ , $L\otimes g^{-1}W$ are simple $A*H_L$ -modules by the above, we conclude that we have an isomorphism

$$ \begin{align*} \varphi":L\otimes V\cong L\otimes g^{-1}W \end{align*} $$

of $A*H_L$ -modules. Now note that

$$ \begin{align*} \operatorname{\mathrm{Hom}}_{A*H_L}(L\otimes V, L\otimes g^{-1}W) &\cong \operatorname{\mathrm{Hom}}_{A}(L\otimes V, L\otimes g^{-1}W)^{H_L}\\ \cong (\operatorname{\mathrm{Hom}}_A(L, L)\otimes \operatorname{\mathrm{Hom}}_{\operatorname{\mathrm{k}}}(V, g^{-1}W))^{H_L} &\cong (\operatorname{\mathrm{Hom}}_{\operatorname{\mathrm{k}}}(V, g^{-1}W))^{H_L} \cong \operatorname{\mathrm{Hom}}_{\operatorname{\mathrm{k}} H_L}(V, g^{-1}W). \end{align*} $$

Hence, $gV\cong W$ .

The following is a counterexample to Lemma 2 in [21], which claims that every simple $A*G$ -module is isomorphic to a module of the form $S\otimes V$ where S is a simple $A*G$ submodule of the socle of A and V is an irreducible representation of G. The error in their proof lies in the erroneous assumption that if S is a simple $A*G$ submodule of the socle of A, then $\operatorname {\mathrm {Hom}}_A(S, S)\cong \operatorname {\mathrm {k}}$ .

While this is in general false, it holds if, for example, all simple $\operatorname {\mathrm {k}} G$ -modules are one-dimensional, that is, if G is commutative. The reason for this is that in this case we have for any simple $A*G$ -module S and any irreducible representation V of G isomorphisms

$$ \begin{align*} &\operatorname{\mathrm{End}}_{A*G}(S\otimes V)\cong \operatorname{\mathrm{End}}_{A}(S\otimes V)^G\cong (\operatorname{\mathrm{End}}_A(S)\otimes \operatorname{\mathrm{End}}_{\operatorname{\mathrm{k}}}(V))^G\\ &\cong (\operatorname{\mathrm{End}}_A(S)\otimes \operatorname{\mathrm{k}})^G\cong \operatorname{\mathrm{End}}_{A*G}(S)\cong \operatorname{\mathrm{k}}, \end{align*} $$

where G acts trivially on $\operatorname {\mathrm {End}}_{\operatorname {\mathrm {k}}}(V)$ as $\operatorname {\mathrm {End}}_{\operatorname {\mathrm {k}}}(V)\cong \operatorname {\mathrm {k}}\cong \operatorname {\mathrm {End}}_{\operatorname {\mathrm {k}} G}(V)=\operatorname {\mathrm {End}}_{\operatorname {\mathrm {k}}}(V)^G$ , so that $S\otimes V$ is irreducible. Thus, the result in [21, Lem. 2] holds in particular if G is commutative.

Example 2.17. Consider $\operatorname {\mathrm {k}}=\mathbb {C}$ , $A:=\operatorname {\mathrm {k}}^5$ , $G:=S_5$ acting on A via permutations of the entries. A is semisimple and basic, and $H_1:=H_{L_1}\cong S_4$ , where $L_1$ is the simple corresponding to the first copy of $\operatorname {\mathrm {k}}$ , so that by the above proposition we have a simple $A*G$ -module $\operatorname {\mathrm {k}} S_5\otimes _{\operatorname {\mathrm {k}} S_{4}}(\operatorname {\mathrm {k}}\times \{0\}^4\otimes V)$ for every irreducible representation V of $S_4$ . In particular, since $S_4$ has an irreducible representation V of dimension 3, $A*G$ has a simple module of dimension

$$ \begin{align*} \dim_{\operatorname{\mathrm{k}}}(\operatorname{\mathrm{k}} S_5\otimes_{\operatorname{\mathrm{k}} S_{4}}(\operatorname{\mathrm{k}}\times \{0\}^4\otimes V))=\dim_{\operatorname{\mathrm{k}}}(\operatorname{\mathrm{k}} S_5/S_{4})\dim_{\operatorname{\mathrm{k}}}V=5\cdot 3=15. \end{align*} $$

Moreover, note that A is a simple $A*G$ -module, and if W is an irreducible representation of $S_5$ ,

$$ \begin{align*} \dim_{\operatorname{\mathrm{k}}}(A\otimes W)=5\dim_{\operatorname{\mathrm{k}}}W. \end{align*} $$

Since $S_5$ has no irreducible representations of dimension 3, this implies that not all simple $A*G$ -modules are of the form $A\otimes W$ . Additionally, note that

$$ \begin{align*} \operatorname{\mathrm{Hom}}_A(A, A)\cong A\ncong \operatorname{\mathrm{k}}. \end{align*} $$

Corollary 2.18. The indecomposable projective $A*G$ -modules are exactly of the form

$$ \begin{align*}\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(P_L\otimes V)\end{align*} $$

for some irreducible $\operatorname {\mathrm {k}} H_L$ -module V. They are isomorphic if and only if there is $g\in G$ such that $gL=L', gV=W$ .

Proof. Since $V|\operatorname {\mathrm {k}} H_L$ , $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(P_L\otimes V)$ is a direct summand of $\operatorname {\mathrm {k}} G\otimes P_L$ as above, and is therefore projective. Moreover, by Proposition 2.12(8),

$$ \begin{align*} \operatorname{\mathrm{top}}(\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(P_L\otimes V))&=\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}\operatorname{\mathrm{top}}(P_L\otimes V)\\ &=\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(\operatorname{\mathrm{top}}(P_L)\otimes V)\\ &=\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(L\otimes V). \end{align*} $$

By Proposition 2.16, this is simple, so that $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(P_L\otimes V)$ is indecomposable.

For any finite-dimensional algebra, we have a bijection between the isomorphism classes of projective indecomposable modules and the isomorphism classes of simple modules given by

$$ \begin{align*} [P]\mapsto [\operatorname{\mathrm{top}}(P)]. \end{align*} $$

Now since by Proposition 2.16 every simple $A*G$ -module is isomorphic to a module of the form $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(L\otimes V)$ , and $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(P_L\otimes V)$ is a projective indecomposable with top $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(L\otimes V)$ , every projective indecomposable $A*G$ -module is isomorphic to a module of the form $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(L\otimes V)$ . Moreover, since two simple $A*G$ -modules $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(L\otimes V)$ and $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_{L'}}(L'\otimes V')$ are isomorphic if and only if there is $g\in G$ such that $gL=L', gV=W$ , their projective covers $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(P_L\otimes V)$ and $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_{L'}}(P_{L'}\otimes V')$ are also isomorphic if and only if there is $g\in G$ such that $gL=L', gV=W$ .

3 Quasi-hereditary algebras

In this section, we shall repeat some standard definitions and results about quasi-hereditary algebras, as introduced by [10], [25]. For an introduction to quasi-hereditary algebras, see, for example, [14].

Let A be an algebra. Denote by $\operatorname {\mathrm {Sim}}(A)$ the set of simple A-modules and suppose $\leq $ is a partial order on $\operatorname {\mathrm {Sim}}(A)$ .

Remark 3.1. Suppose $\leq $ is a partial order on the set $\operatorname {\mathrm {Sim}}(A)$ of simple A-modules. Then this induces a partial order on its additive closure $\operatorname {\mathrm {add}}(\operatorname {\mathrm {Sim}}(A))$ via

$$ \begin{align*} S\leq S'\Leftrightarrow L\leq L' \text{ for all } L|S, L'|S'. \end{align*} $$

Thus, if $\leq $ is a partial order on $\operatorname {\mathrm {Sim}}(A)$ , we will also use it to compare semisimple modules.

Definition 3.2. [14, p. 3]

We call a partial order $\leq $ on $\operatorname {\mathrm {Sim}}(A)$ adapted, if all ${M\in \operatorname {\mathrm {mod}} A}$ with simple top $\operatorname {\mathrm {top}}(M)=L$ and simple socle $\operatorname {\mathrm {soc}}(M)=L'$ , such that L and $L'$ are incomparable with respect to $\leq $ , have a composition factor $L"$ such that $L">L$ and $L">L'$ .

Lemma 3.3.

1. 1. Let M be a module with a composition factor $L'$ . Then there is a factor module $M'$ of M with socle $\operatorname {\mathrm {soc}}(M')\cong L'$ .

2. 2. Let M be a module with a composition factor $L'$ . Then there is a submodule $M'$ of M with top $\operatorname {\mathrm {top}}(M')\cong L'$ .

Proof.

1. 1. By definition of a composition factor, there is a factor module $M"$ of M such that we have an embedding $\iota :L'\rightarrow M"$ .

   Now let N be a maximal submodule of $M"$ subject to $N\cap \operatorname {\mathrm {im}}(\iota )=(0)$ . Then ${M':=M"/N}$ is also a factor module of M. Denote by

   $$ \begin{align*}\pi:M"\rightarrow M'\end{align*} $$
   the canonical projection. Then $\pi \circ \iota :L'\rightarrow M'$ is injective, so that $L'|\operatorname {\mathrm {soc}}(M')$ . Write $\operatorname {\mathrm {soc}}(M')=\pi \circ \iota (L')\oplus S$ . Then $N\subseteq \pi ^{-1}(S)$ and $\iota (L')\cap \pi ^{-1}(S)=(0)$ . Hence, by maximality of N, $\pi ^{-1}(S)=N$ , so that $S=(0)$ .

2. 2. This is dual to 1.

Lemma 3.4. The partial order $\leq $ is adapted if and only if every module M which has a composition factor $L'$ such that $L'$ is incomparable to every summand L of its top $\operatorname {\mathrm {top}}(M)$ has a composition factor $L"$ and a composition factor $L|\operatorname {\mathrm {top}}(M)$ which is a summand of the top, such that $L">L$ .

Proof. Suppose $\leq $ is adapted, and let M be a module with a composition factor $L'$ such that $L'$ is incomparable to every summand L of its top $\operatorname {\mathrm {top}}(M)$ .

By Lemma 3.3, M has a factor module $M'$ with simple socle $L'$ . Let L be any summand of $\operatorname {\mathrm {top}}(M')$ . Then, since $M'$ is a factor module of M, we have $\operatorname {\mathrm {top}}(M')|\operatorname {\mathrm {top}}(M)$ . Hence, L is also a summand of $\operatorname {\mathrm {top}}(M)$ , and thus in particular incomparable to $L'$ . Moreover, we can again apply Lemma 3.3 to obtain a submodule $M"$ of $M'$ with simple top $\operatorname {\mathrm {top}}(M")\cong L$ . As $M"$ is a submodule of $M'$ , $\operatorname {\mathrm {soc}}(M")|\operatorname {\mathrm {soc}}(M')\cong L'$ , so that $M"$ has simple socle isomorphic to $L'$ . Since L and $L'$ are incomparable, $M"$ thus has a composition factor $L">L, L'$ , and since $L"$ is a composition factor of $M"$ , which is a submodule of a factor module of M, $L"$ is also a composition factor of M. Hence, this proves the first implication.

On the other hand, suppose that every module M which has a composition factor $L'$ such that $L'$ is incomparable to every summand of its top has a composition factor $L"$ and a composition factor $L|\operatorname {\mathrm {top}}(M)$ such that $L">L$ , and let M be a module with simple top L and simple socle $L'$ such that L and $L'$ are incomparable.

Then, by assumption, M has a composition factor $L"$ such that $L">L$ . Without loss of generality, we can choose $L"$ maximal with respect to $L">L$ . Then, by Lemma 3.3, M has a submodule $M'$ with simple top $\operatorname {\mathrm {top}}(M')\cong L"$ . Since M has simple socle $L'$ , $M'$ also has simple socle $L'$ . Now, if $L'$ and $L"$ were incomparable, then by assumption $M'$ would have a composition factor $L"'>L"$ , which is a contradiction to the maximality of $L"$ . Hence, $L'$ and $L"$ are comparable. Since $L">L$ and $L'$ and L are incomparable, this implies that $L'<L"$ .

Definition 3.5. Let $\leq $ be a partial order on $\operatorname {\mathrm {Sim}}(A)$ . Then, for every simple A-module L, we define

$$ \begin{align*} \Delta_L:=P_L/\left(\sum_{L'\nleq L, \varphi\in \operatorname{\mathrm{Hom}}_A(P_{L'}, P_L)}\operatorname{\mathrm{im}}(\varphi)\right) \end{align*} $$

and

$$ \begin{align*} \widehat{\Delta}_L:=P_L/\left(\sum_{L'> L, \varphi\in \operatorname{\mathrm{Hom}}_A(P_{L'}, P_L)}\operatorname{\mathrm{im}}(\varphi)\right). \end{align*} $$

Denote by $\pi _L:P_L\rightarrow \Delta _L$ and $\widehat {\pi }_L:P_L\rightarrow \widehat {\Delta }_L'$ the canonical projection. Moreover, write

$$ \begin{align*}\Delta:=\bigoplus_{L\in \operatorname{\mathrm{Sim}}(A)}\Delta_L\end{align*} $$

and

$$ \begin{align*}\widehat{\Delta}:=\bigoplus_{L\in \operatorname{\mathrm{Sim}}(A)}\widehat{\Delta}_L\end{align*} $$

and call $(\Delta _L)_{L\in \operatorname {\mathrm {Sim}}(A)}$ the collection of standard modules and $(\widehat {\Delta }_L)_{L\in \operatorname {\mathrm {Sim}}(A)}$ the collection of pseudostandard modules for $(A, \leq )$ .

Later, in case more than one algebra is involved, we will sometimes add a superscript to $\Delta $ , $\widehat {\Delta }$ , $\Delta _L$ , and $\widehat {\Delta }_L$ indicating the respective algebra.

Lemma 3.6. Let $L\in \operatorname {\mathrm {Sim}}(A)$ . Then the following statements hold:

1. 1. $L'\nleq L$ for every summand $L'$ of $\operatorname {\mathrm {top}}(\ker (\pi _L))$ and for every epimorphism $f:P_L\rightarrow M$ such that $L'\nleq L$ for every summand $L'$ of $\operatorname {\mathrm {top}}(\ker (f))$ we have an epimorphism ${g:M\rightarrow \Delta _L}$ such that $\pi _L=g\circ f$ .

2. 2. $L' \leq L$ for every composition factor $L'$ of $\Delta _L$ and for every homomorphism $f:P_L\rightarrow M$ such that $L'\leq L$ for every composition factor $L'$ of M we have a homomorphism ${g:\Delta _L\rightarrow M}$ such that $f=g\circ \pi _L$ .

3. 3. We have ${\operatorname {\mathrm {top}}(\ker (\widehat {\pi }_L))>L}$ and for every epimorphism ${f:P_L\rightarrow M}$ such that $\operatorname {\mathrm {top}}(\ker (f))>L$ we have an epimorphism $g:M\rightarrow \widehat {\Delta }_L$ such that $\widehat {\pi }_L=g\circ f$ .

4. 4. $L' \ngtr L$ for every composition factor $L'$ of $\widehat {\Delta }_L'$ and for every homomorphism $f:P_L\rightarrow M$ such that $L'\ngtr L$ for every composition factor $L'$ of M we have a homomorphism ${g:\widehat {\Delta }_L'\rightarrow M}$ such that $f=g\circ \widehat {\pi }_L$ .

Proof.

1. 1. We have $L'\nleq L$ for every summand $L'$ of $\operatorname {\mathrm {top}}(\ker (\pi _L))$ by definition. So let $f:P_L\rightarrow M$ be an epimorphism such that $L'\nleq L$ for every summand $L'$ of $\operatorname {\mathrm {top}}(\ker (f))$ . Then we have a projection $\pi :\bigoplus _{L'\nleq L}n_{L'}P_{L'}\rightarrow \ker (f)$ for some $n_{L'}\in \mathbb {N}_0$ . Composing with the embedding yields that

   $$ \begin{align*} \ker(f)\subseteq \sum_{L'\nleq L, \varphi\in \operatorname{\mathrm{Hom}}_A(P_{L'}, P_L)}\operatorname{\mathrm{im}}(\varphi)=\ker(\pi_L). \end{align*} $$
   Hence, $\pi _L$ factors through f.

2. 2. By definition, $L'\leq L$ for every composition factor $L'$ of $\Delta _L$ . So let $f:P_L\rightarrow M$ such that $L'\nleq L$ for every composition factor $L'$ of M. Let $L'\nleq L$ and let $\varphi :P_{L'}\rightarrow P_L$ . Then, since all composition factors of M are less than or equal to L, $f\circ \varphi =0$ . Hence, $\operatorname {\mathrm {im}}(\varphi )\subseteq \ker (f)$ , so that $\ker (\pi _L)\subseteq \ker (f)$ and thus f factors through $\pi _L$ .

3. 3. This is analogous to 1.

4. 4. This is analogous to 2.

Lemma 3.7. For every $L, L'\in \operatorname {\mathrm {Sim}}(A)$ such that $\operatorname {\mathrm {Ext}}^1(\widehat {\Delta }_{L'}, \Delta _{L})\neq (0)$ we have $L'<L$

Proof. By definition, the module $\widehat {\Delta }_{L'}$ has a projective presentation

for some integers $n_{L"}\in \mathbb {N}_0$ . Suppose $\operatorname {\mathrm {Ext}}^1_A(\widehat {\Delta }_{L'}, \Delta _{L})\neq (0)$ . Then

$$ \begin{align*} \operatorname{\mathrm{Hom}}_A({\bigoplus_{L">L'}n_{L"}P_{L"}}, \Delta_L)\neq (0), \end{align*} $$

so that for some $L">L'$ ,

$$ \begin{align*} \operatorname{\mathrm{Hom}}_A({P_{L"}}, \Delta_L)\neq (0). \end{align*} $$

Thus, $L"$ is a composition factor of $\Delta _L$ . Now since every composition factor of $\Delta _{L}$ is less than or equal to L, this implies $L' < L"\leq L$ .

The following definition is due to [14]; it resembles the definition of an exceptional collection, originating in the work of Beilinson [1], [2], developed in [3], [16], the only difference being that condition 3. is here only required for $\operatorname {\mathrm {Ext}}^1$ instead of for $\operatorname {\mathrm {Ext}}^n$ for all $n\geq 1$ .

Definition 3.8. Let $\leq $ be a partial order on $\operatorname {\mathrm {Sim}}(A)$ . Then a standardizable set for $(A, \leq )$ is a family of modules $M=(M_L)_{L\in \operatorname {\mathrm {Sim}}(A)}$ such that:

1. 1. $\operatorname {\mathrm {top}}(M_L)\cong L$ .

2. 2. $\operatorname {\mathrm {Hom}}_A(M_L, M_{L'})\neq (0)\Rightarrow L\leq L'$ .

3. 3. $\operatorname {\mathrm {Ext}}_A^1(M_L, M_{L'})\neq (0)\Rightarrow L< L'$ .

Remark 3.9. Note that $(\Delta _L)_L$ fulfills conditions 1. and 2. in the above definition. Thus, by Lemma 3.7, if $\widehat {\Delta }_L=\Delta _L$ for every $L\in \operatorname {\mathrm {Sim}}(A)$ , then $(\Delta _L)_L=(\widehat {\Delta }_{L})_L$ is a standardizable set.

The following lemma tells us that $(\Delta _L)_L=(\widehat {\Delta }_{L})_L$ if and only if $\leq $ is adapted. Moreover, the former is the case if and only if any refinement of our partial order will give rise to the same set of standard modules. Thus, being adapted means that our partial order is, in a sense, fine enough.

Lemma 3.10. The following statements are equivalent:

1. 1. $\Delta _L=\widehat {\Delta }_L$ for every $L\in \operatorname {\mathrm {Sim}}(A)$ .

2. 2. $\leq $ is adapted.

3. 3. $(\widehat {\Delta }_L)_L$ is a standardizable set.

4. 4. $\operatorname {\mathrm {Hom}}_A(\widehat {\Delta }_{L'}, \widehat {\Delta }_L)\neq (0) \Rightarrow L'\leq L$ .

Proof.

1. 1 ⇒ 2 Suppose $\Delta _L=\widehat {\Delta }_L$ for every $L\in \operatorname {\mathrm {Sim}}(A)$ , and let M be an A-module with simple top L and socle $L'$ . Suppose that no composition factor $L"$ of M is bigger than L. Since $L=\operatorname {\mathrm {top}}(M)$ , there is an epimorphism $\pi _M:P_L\rightarrow M$ . Now since no composition factor $L"$ of M is bigger than L, Lemma 3.6 yields a homomorphism $g:\widehat {\Delta }_L\rightarrow M$ such that $\pi _M=g\circ \widehat {\pi }_L$ . In particular, g is an epimorphism, so that, since every composition factor of $\widehat {\Delta }_L=\Delta _L$ is less than or equal to L, every composition factor of M is less than or equal to L. In particular, $L'$ and L are comparable.

2. 2 ⇒ 3 This holds by Remark 3.9.

3. 3 ⇒ 4 This holds by definition.

4. 4 ⇒ 1 Suppose that there is some $L\in \operatorname {\mathrm {Sim}}(A)$ such that $\Delta _L\neq \widehat {\Delta }_L$ . Then $\widehat {\Delta }_L$ has some composition factor $L'\nleq L$ . Let $L'$ be a maximal such composition factor. Then there is a nonzero homomorphism $f:P_{L'}\rightarrow \widehat {\Delta }_L$ . Moreover, since $L'$ is a maximal composition factor of $\widehat {\Delta }_L$ , we have $L"\ngtr L'$ for every composition factor $L"$ of $\widehat {\Delta }_L$ . Thus, Lemma 3.6 yields a homomorphism $g: \widehat {\Delta }_{L'}\rightarrow \widehat {\Delta }_L$ such that $f=g\circ \widehat {\pi }_{L'}$ . In particular,

   $$ \begin{align*} \operatorname{\mathrm{Hom}}_A(\widehat{\Delta}_{L'}, \widehat{\Delta}_L)\neq (0).\\[-39pt] \end{align*} $$

Example 3.11. Consider the algebra A given by the quiver

with relations $\langle \beta \alpha \rangle =J^2$ and the partial order on $\operatorname {\mathrm {Sim}}(A)=\{L_a, L_b, L_c\}$ given by $L_c<L_a$ and $L_c<L_b$ . Then the indecomposable projective modules are given by

$$ \begin{align*} P_a=\begin{pmatrix} a \\ b\end{pmatrix}\text{, } P_b=\begin{pmatrix} b \\ c\end{pmatrix} \text{ and } P_c=(c), \end{align*} $$

so $\widehat {\Delta }_i=P_i$ for every $i\in \{a, b, c\}$ . In particular, $\operatorname {\mathrm {Hom}}_A(\widehat {\Delta }_b, \widehat {\Delta }_a)\neq (0)$ , so that $(\widehat {\Delta }_i)_i$ is not standardizable.

On the other hand, $\Delta _a=L_a$ , $\Delta _b=P_b$ , and $\Delta _c=P_c$ , so that

$$ \begin{align*} \operatorname{\mathrm{Ext}}^1_A(\Delta_j, \Delta_i)=(0) \end{align*} $$

for all $n\geq 1$ , $i\in \{a,b,c\}$ and $j\in \{b,c\}$ . Moreover,

is a projective resolution of $L_a$ , where $r_\alpha $ and $r_\beta $ denote right multiplication by $\alpha $ (resp. $\beta $ ) and $\pi _a$ is the canonical projection. Since $\operatorname {\mathrm {Hom}}_A(P_b, \Delta _a)=\operatorname {\mathrm {Hom}}_A(P_b, \Delta _c)=(0)$ , this implies that

$$ \begin{align*} \operatorname{\mathrm{Ext}}^1_A(\Delta_a, \Delta_a)=(0)=\operatorname{\mathrm{Ext}}^1_A(\Delta_a, \Delta_c). \end{align*} $$

Finally, since

$$ \begin{align*} r_\beta^*:\operatorname{\mathrm{Hom}}_A(P_b, \Delta_b)\rightarrow \operatorname{\mathrm{Hom}}_A(P_c, \Delta_b) \end{align*} $$

is injective, we also obtain that $\operatorname {\mathrm {Ext}}^1_A(\Delta _a, \Delta _b)=(0)$ . Thus, $(\Delta _i)_i$ is standardizable.

The following definition is an adaptation of the definition of an exceptional sheaf which can be found in [16], where, as in Definition 3.8, we replace the requirement on $\operatorname {\mathrm {Ext}}^n$ for $n\geq 0$ by a requirement only on $\operatorname {\mathrm {Ext}}^1$ .

Definition 3.12. An A-module M is called exceptional if $\operatorname {\mathrm {Ext}}_A^1(M, M)=(0)$ and $\operatorname {\mathrm {End}}_A(M)\cong \operatorname {\mathrm {k}}$ .

Lemma 3.13. The following statements are equivalent:

1. 1. $\leq $ is adapted and $\Delta _L=\widehat {\Delta }_L$ is exceptional for every $L\in \operatorname {\mathrm {Sim}}(A)$ .

2. 2. Every composition factor $L'$ of $\operatorname {\mathrm {rad}}(\widehat {\Delta }_{L})$ fulfills $L'<L$ .

3. 3. $\operatorname {\mathrm {Hom}}_A(\widehat {\Delta }_{L'}, \operatorname {\mathrm {rad}}(\widehat {\Delta }_{L}))\neq (0)\Rightarrow L'<L$ .

Proof.

1. 1 ⇒ 2 Recall that if $\leq $ is adapted, then $(\Delta _L)_L=(\widehat {\Delta }_L)_L$ . Let $L'$ be a composition factor of $\operatorname {\mathrm {rad}}(\widehat {\Delta }_{L})=\operatorname {\mathrm {rad}}(\Delta _L)$ . Then $L'\leq L$ . If $L=L'$ , then this induces a nonzero homomorphism $P_{L}\rightarrow \operatorname {\mathrm {rad}}(\Delta _{L})$ and thus, by Lemma 3.6, an endomorphism $\Delta _{L}\rightarrow \operatorname {\mathrm {rad}}(\Delta _{L})\rightarrow \Delta _{L}$ which is neither zero nor invertible. This is a contradiction. Thus, $L'<L$ .

2. 2 ⇒ 3 Suppose $\operatorname {\mathrm {Hom}}_A(\widehat {\Delta }_{L'}, \operatorname {\mathrm {rad}}(\widehat {\Delta }_{L}))\neq (0)$ . Then, since $\widehat {\Delta }_{L'}$ has simple top $L'$ , $L'$ is a composition factor of $\operatorname {\mathrm {rad}}(\widehat {\Delta }_{L})$ and thus $L'<L$ .

3. 3 ⇒ 1 If $L\neq L'$ , then

   $$ \begin{align*} \operatorname{\mathrm{Hom}}_A(\widehat{\Delta}_{L'}, \operatorname{\mathrm{rad}}(\widehat{\Delta}_{L}))= (0)\Leftrightarrow \operatorname{\mathrm{Hom}}_A(\widehat{\Delta}_{L'}, \widehat{\Delta}_{L})=(0). \end{align*} $$

   Thus,

   $$ \begin{align*} \operatorname{\mathrm{Hom}}_A(\widehat{\Delta}_{L'}, \widehat{\Delta}_{L})\neq (0)\Rightarrow L'\leq L, \end{align*} $$

   so that $\leq $ is adapted, $\widehat {\Delta }_L=\Delta _L$ , and $(\Delta _L)_L$ is a standardizable set by Lemma 3.10.

   In particular, $\operatorname {\mathrm {Ext}}^1(\Delta _L, \Delta _L)=(0).$

   Moreover, $\operatorname {\mathrm {Hom}}_A(\widehat {\Delta }_L, \operatorname {\mathrm {rad}}(\widehat {\Delta }_L))=(0)$ , so that any endomorphism of $\widehat {\Delta }_L$ is either surjective or zero. Thus, $\operatorname {\mathrm {End}}_A(\Delta _L)=\operatorname {\mathrm {End}}_A(\widehat {\Delta }_L)\cong \operatorname {\mathrm {k}}.$

Definition 3.14. We denote by $\mathcal {F}(\Delta )$ the full subcategory of $\operatorname {\mathrm {mod}} A$ which contains all A-modules admitting a filtration by the $\Delta _L$ , $L\in \operatorname {\mathrm {Sim}}(A)$ . In other words, an A-module M is in $\mathcal {F}(\Delta )$ if and only if there is an integer $m\geq 0$ and an ascending sequence of submodules

$$ \begin{align*} (0)=M_0\subset M_1 \subset \dots \subset M_m=M \end{align*} $$

such that for every $1\leq i\leq m$ there is an $L_i\in \operatorname {\mathrm {Sim}}(A)$ such that $M_i/M_{i-1}\cong \Delta _{L_i}$ .

Proposition 3.15. [14, Lems. 1.4 and 1.5]

The subcategory $\mathcal {F}(\Delta )$ is closed under direct sums, direct summands, and extensions.

Definition 3.16. [10], [12], [24]

An algebra A together with an adapted partial order $\leq $ on $\operatorname {\mathrm {Sim}}(A)$ is called:

1. 1. left standardly stratified, if $A\in \mathcal {F}(\Delta );$

2. 2. quasi-hereditary, if additionally $\Delta _L$ is exceptional for all $L\in \operatorname {\mathrm {Sim}}(A)$ ;

3. 3. strongly quasi-hereditary, if additionally every $\Delta _L$ has projective dimension 1;

4. 4. directed, if $\Delta _L\cong L$ for all $L\in \operatorname {\mathrm {Sim}}(A)$ .

Definition 3.17. ([17, p. 405], [4, Def. 3.4])

Let $(A, \leq )$ be a quasi-hereditary algebra. Then a subalgebra $B\subseteq A$ is called an exact Borel subalgebra if there is a bijection $i:\operatorname {\mathrm {Sim}}(B)\rightarrow \operatorname {\mathrm {Sim}}(A)$ such that:

1. 1. A is projective as a right B-module;

2. 2. B is directed;

3. 3. $A\otimes _B L=\Delta _{i(L)}$ for all $L\in \operatorname {\mathrm {Sim}}(B)$ .

Moreover, it is called:

1. 1. a strong exact Borel subalgebra if there is a maximal semisimple subalgebra of A which is also a semisimple subalgebra of B;

2. 2. a homological exact Borel subalgebra if the induced maps

   $$ \begin{align*} A\otimes_B- :\operatorname{\mathrm{Ext}}_B^*(L, L')\rightarrow \operatorname{\mathrm{Ext}}_A^*(\Delta_L, \Delta_{L'}) \end{align*} $$
   are isomorphisms in degree greater than or equal to 2 and epimorphisms in degree 1 for all $L, L'\in \operatorname {\mathrm {Sim}}(B)$ ;

3. 3. a normal exact Borel subalgebra, if there is a splitting of the inclusion $\iota :B\rightarrow A$ whose kernel is a right ideal in A; and

4. 4. a regular exact Borel subalgebra if it is normal and the induced maps

   $$ \begin{align*} A\otimes_B- :\operatorname{\mathrm{Ext}}_B^*(L, L')\rightarrow \operatorname{\mathrm{Ext}}_A^*(\Delta_{i(L)}, \Delta_{i(L')}) \end{align*} $$
   are isomorphisms in degree greater than or equal to 1 for all $L, L'\in \operatorname {\mathrm {Sim}}(B)$ .

Remark 3.18. Let A be a finite-dimensional algebra, $L^{(A)}:=A/\operatorname {\mathrm {rad}}(A)$ , and $\pi _A:A\rightarrow A/\operatorname {\mathrm {rad}}(A)$ be the canonical projection. By definition, $L^{(A)}$ is the maximal semisimple quotient of A. However, recall that by the Wedderburn–Malcev theorem, there is an embedding

$$ \begin{align*}\iota_A:L^{(A)}\rightarrow A\end{align*} $$

such that $\pi _A\circ \iota _A=\operatorname {\mathrm {id}}_{L^{(A)}}$ , which turns $L^{(A)}$ into a maximal semisimple subalgebra of A. Moreover, by the same result, any two maximal semisimple subalgebras of A are conjugated, so that, in particular, $L^{(A)}$ is up to isomorphism the unique semisimple subalgebra of A.

Lemma 3.19. Suppose that B is an exact Borel subalgebra of A. Then B is a strong exact Borel subalgebra of A if and only if $A\operatorname {\mathrm {rad}}(B)\subseteq \operatorname {\mathrm {rad}}(A)$ .

Proof. Let $L^{(A)}:=A/\operatorname {\mathrm {rad}}(A)$ and $L^{(B)}:=B/\operatorname {\mathrm {rad}}(B)$ . By Remark 3.18, $L^{(A)}$ and $L^{(B)}$ are up to isomorphism the unique maximal semisimple subalgebras of A and B, respectively.

In particular, a semisimple subalgebra of A is a maximal semisimple subalgebra if and only if it has the same vector space dimension as $L^{(A)}$ . Since any semisimple subalgebra of B is also a semisimple subalgebra of A, this means that B contains a maximal semisimple subalgebra of A if and only if $\dim _{\operatorname {\mathrm {k}}}L^{(A)}=\dim _{\operatorname {\mathrm {k}}}L^{(B)}$ .

Thus, B is a strong exact Borel subalgebra of A if and only if $\dim _{\operatorname {\mathrm {k}}}L^{(B)}=\dim _{\operatorname {\mathrm {k}}}L^{(A)}$ .

Let $L_1^B, \dots , L_n^B$ be a set of representative of the simple B-modules. Write $\Delta _i^A:=A\otimes _B L_i^B$ and $L_i^A:=\operatorname {\mathrm {top}}(A\otimes _B L_i^B)$ . Since B is an exact Borel subalgebra of A, $L_1^A, \dots , L_n^A$ is a set of representatives of the simple A-modules, and $\Delta _1^A, \dots , \Delta _n^A$ are the corresponding standard modules.

Let $X:=A\operatorname {\mathrm {rad}}(B)/(\operatorname {\mathrm {rad}}(A)\cap A\operatorname {\mathrm {rad}}(B))$ . Then, as A-modules,

$$ \begin{align*} L^{(A)}= A/\operatorname{\mathrm{rad}}(A) &\cong ((\operatorname{\mathrm{rad}}(A)+A\operatorname{\mathrm{rad}}(B))/\operatorname{\mathrm{rad}}(A))\oplus A/(\operatorname{\mathrm{rad}}(A)+A\operatorname{\mathrm{rad}}(B))\\ &\cong A\operatorname{\mathrm{rad}}(B)/(\operatorname{\mathrm{rad}}(A)\cap A\operatorname{\mathrm{rad}}(B))\oplus \operatorname{\mathrm{top}}(A/A\operatorname{\mathrm{rad}}(B))\\ &\cong X\oplus \operatorname{\mathrm{top}}\left(A\otimes_B B/\operatorname{\mathrm{rad}}(B)\right)\\ &\cong X\oplus \operatorname{\mathrm{top}}\left(\bigoplus_{i}[\operatorname{\mathrm{top}} B: L_i^B]\Delta_{L_i^A}\right)\\ &\cong X\oplus \bigoplus_{i}[L^{(B)}: L_i^B]L_i^A. \end{align*} $$

Recall that for any $n\in \mathbb {N}$ we have $\dim _{\operatorname {\mathrm {k}}}\operatorname {\mathrm {k}}^n=n=[\operatorname {\mathrm {Mat}}_n(\operatorname {\mathrm {k}}):\operatorname {\mathrm {k}}^n]$ . Moreover, by the Wedderburn–Artin theorem, any finite-dimensional semisimple algebra over $\operatorname {\mathrm {k}}$ is isomorphic to a direct sum of matrix rings, and for any two finite-dimensional algebras R and $R'$ , $\operatorname {\mathrm {Sim}}(R\oplus R')$ is the disjoint union of $\operatorname {\mathrm {Sim}}(R)$ and $\operatorname {\mathrm {Sim}}(R')$ where R acts on $L'\in \operatorname {\mathrm {Sim}}(R')$ via zero and analogously for $L\in \operatorname {\mathrm {Sim}}(R)$ . Hence, the equation above generalizes to any finite-dimensional semisimple algebra over $\operatorname {\mathrm {k}}$ , so that we have

$$ \begin{align*} \dim_{\operatorname{\mathrm{k}}}L_i^A&=[L^{(A)}:L_i^A] \end{align*} $$

for every $1\leq i\leq n$ . Thus,

$$ \begin{align*} \dim_{\operatorname{\mathrm{k}}}L_i^A&=[L^{(A)}:L_i^A]=[X:L_i^A]+[L^{(B)}: L_i^B]=[X:L_i^A]+\dim_{\operatorname{\mathrm{k}}}L_i^B, \end{align*} $$

so that

$$ \begin{align*} \dim_{\operatorname{\mathrm{k}}} L^{(A)}&=\dim_{\operatorname{\mathrm{k}}} X+\sum_i[L^{(B)}: L_i^B]\dim_{\operatorname{\mathrm{k}}} L_i^A\\ &= \dim_{\operatorname{\mathrm{k}}} X+\sum_i[L^{(B)}: L_i^B]([X:L_i^A]+\dim_{\operatorname{\mathrm{k}}}(L_i^B))\\ &=\dim_{\operatorname{\mathrm{k}}} X+\sum_{i}[X:L_i^A][L^{(B)}:L_i^B]+ \dim_{\operatorname{\mathrm{k}}}L^B. \end{align*} $$

Hence, B is a strong exact Borel subalgebra of A if and only if $X=(0)$ , that is, if $A\operatorname {\mathrm {rad}}(B)\subseteq B$ .

4 Quasi-hereditary algebras with a group action

Throughout, let, as before, A be a finite-dimensional $\operatorname {\mathrm {k}}$ -algebra and let G be a group acting on A via automorphisms such that the order of G does not divide the characteristic of $\operatorname {\mathrm {k}}$ .

Definition 4.1. A partial order $\leq _A$ on $\operatorname {\mathrm {Sim}}(A)$ is called G-equivariant if

$$ \begin{align*} L<_AL' \Leftrightarrow gL<_AhL' \text{ for all } g,h\in G. \end{align*} $$

On the other hand, a partial order $\leq _{A*G}$ on $\operatorname {\mathrm {Sim}}(A*G)$ is called G-stable if

$$ \begin{align*} S<_{A*G}S' \Leftrightarrow S\otimes V<_{A*G} S' \otimes W \end{align*} $$

for all $\operatorname {\mathrm {k}} G$ -modules $V, W$ .

Definition 4.2.

1. 1. Let $\leq $ be a G-equivariant partial order on $\operatorname {\mathrm {Sim}}(A)$ . Then we define a partial order $\leq _{G}$ on $\operatorname {\mathrm {Sim}}(A*G)$ via

   $$ \begin{align*} S<_GS':\Leftrightarrow _{A|}S<_{|A}S'. \end{align*} $$

2. 2. Let $\leq '$ be a G-stable partial order on $\operatorname {\mathrm {Sim}}(A*G)$ . Then we define a partial order $\leq ^{\prime }_{|G}$ on $\operatorname {\mathrm {Sim}}(A)$ via

   $$ \begin{align*} L<^{\prime}_{|G}L':\Leftrightarrow \operatorname{\mathrm{k}} G\otimes L <'\operatorname{\mathrm{k}} G\otimes L', \end{align*} $$
   where we extend the partial orders from the simple modules to the semisimple modules as in Remark 3.1.

Hence, we define a strict partial order $(<_{A})_G$ as the pullback of a strict partial order $<_A$ along the map

$$ \begin{align*} \operatorname{\mathrm{add}}{\operatorname{\mathrm{Sim}}(A*G)}\rightarrow \operatorname{\mathrm{add}}(\operatorname{\mathrm{Sim}}(A)), M\mapsto {}_{|A}M, \end{align*} $$

and similarly a strict partial order $(<_{A*G})_{|G}$ as the pullback of a strict partial order $<_{A*G}$ along the map

$$ \begin{align*} \operatorname{\mathrm{add}}{\operatorname{\mathrm{Sim}}(A)}\rightarrow \operatorname{\mathrm{add}}(\operatorname{\mathrm{Sim}}(A*G)), M\mapsto I_GM. \end{align*} $$

Note that since the above maps are not necessarily injective, the pullbacks of $\leq _A$ (resp. $\leq _{A*G}$ ) along these maps are not necessarily partial orders. For example, if we consider the case $A=\operatorname {\mathrm {k}}$ , $G=\mathbb {Z}/2\mathbb {Z}$ with the trivial action, and the unique partial order $\leq _A$ on $\operatorname {\mathrm {Sim}}(A)$ , then we have $A*G\cong \operatorname {\mathrm {k}}^2$ with simple modules $L_1$ and $L_2$ corresponding to the first and second copies of $\operatorname {\mathrm {k}}$ , respectively. In this setting, if we tried to define $L_i\leq _{A*G} L_j\Leftrightarrow R_GL_i\leq _A R_GL_j$ , we would obtain that $L_1\leq L_2$ and $L_2\leq L_1$ since $R_GL_1\cong R_gL_2$ , that is, the asymmetry of $\leq _{A*G}$ would be violated, so that it would not be a partial order.

On the other hand, the pullback of a strict partial order along any map is always a strict partial order.

In particular, the above pullbacks yield well-defined partial orders $\leq _G$ and $\leq ^{\prime }_{|G}$ even for a not necessarily G-equivariant partial order $\leq $ and a not necessarily G-stable partial order  $\leq '$ .

However, if, for example, $\leq $ is not G-equivariant, then there may be some simple $A*G$ modules $S\neq S'$ such that $L< L'$ , but $L"\nless L"'$ for some simple summands of $L, L"$ of ${}_{A|}S$ and $L', L"'$ of ${}_{A|}S'$ , so S and $S'$ would be incomparable with respect to $\leq _G$ , even though all summands of ${}_{|A}S$ and ${}_{|A}S'$ may be comparable. Hence, even a total order $\leq $ might result in an empty order $\leq _G$ , and so there is little hope to conclude adaptedness of $\leq _G$ from adaptedness of $\leq $ .

Similar considerations hold for a non-G-stable partial order $\leq '$ and its induced partial order $<^{\prime }_{|G}$ .

Proposition 4.3.

1. 1. Let $\leq $ be a G-equivariant partial order on $\operatorname {\mathrm {Sim}}(A)$ . Then $\leq _{G}$ is the unique G-stable partial order such that

   $$ \begin{align*} L< L'\Leftrightarrow \operatorname{\mathrm{k}} G\otimes L<_{G}\operatorname{\mathrm{k}} G\otimes L'. \end{align*} $$

2. 2. Let $\leq '$ be a G-stable partial order on $\operatorname {\mathrm {Sim}}(A*G)$ . Then $\leq ^{\prime }_{|G}$ is the unique G-equivariant partial order on $\operatorname {\mathrm {Sim}}(A)$ such that

   $$ \begin{align*} S<'S'\Leftrightarrow _{A|}S(<^{\prime}_{|G})_{A|}S'. \end{align*} $$

Proof.

1. 1. Clearly, $\leq _G$ is G-stable, and

   $$ \begin{align*} L<L'\Leftrightarrow \operatorname{\mathrm{k}} G\otimes L<_{G}\operatorname{\mathrm{k}} G\otimes L'. \end{align*} $$
   Now suppose that $\leq '$ is another G-stable partial order on $\operatorname {\mathrm {Sim}}(A*G)$ such that
   $$ \begin{align*} L<L'\Leftrightarrow \operatorname{\mathrm{k}} G\otimes L<'\operatorname{\mathrm{k}} G\otimes L'. \end{align*} $$
   Let $S, S'\in \operatorname {\mathrm {Sim}}(A*G)$ . Then there are $L, L'\in \operatorname {\mathrm {Sim}}(A)$ such that $L|_{A|}S$ , $L'|_{A|}S'$ . Thus, $\operatorname {\mathrm {k}} G\otimes L| \operatorname {\mathrm {k}} G\otimes {}_{A|}S\cong S\otimes \operatorname {\mathrm {k}} G$ and $\operatorname {\mathrm {k}} G\otimes L'| \operatorname {\mathrm {k}} G\otimes {}_{A|}S'\cong S'\otimes \operatorname {\mathrm {k}} G$ .

   Moreover, by Corollary 2.14, there are $L", L"'\in \operatorname {\mathrm {Sim}}(A)$ such that $S|\operatorname {\mathrm {k}} G\otimes L", S'|\operatorname {\mathrm {k}} G\otimes L"'$ . In particular, $L|{}_{A|}\operatorname {\mathrm {k}} G\otimes L"$ and $L'|{}_{A|}\operatorname {\mathrm {k}} G\otimes L"'$ , so that $L\cong gL"$ and $L'\cong hL"'$ for some $g, h\in G$ . Hence, $\operatorname {\mathrm {k}} G\otimes L\cong \operatorname {\mathrm {k}} G\otimes L"$ and $\operatorname {\mathrm {k}} G\otimes L'\cong \operatorname {\mathrm {k}} G\otimes L"'$ . Thus,

   $$ \begin{align*} S<'S'\Leftrightarrow S\otimes \operatorname{\mathrm{k}} G<'S'\otimes \operatorname{\mathrm{k}} G &\Rightarrow \operatorname{\mathrm{k}} G\otimes L<'\operatorname{\mathrm{k}} G\otimes L'\\ &\Leftrightarrow L<L' \Leftrightarrow \operatorname{\mathrm{k}} G\otimes L<_G\operatorname{\mathrm{k}} G\otimes L' \Rightarrow S<_{G}S', \end{align*} $$
   and analogously $S<_GS'\Rightarrow S<' S'.$

2. 2. Clearly, $<^{\prime }_{|G}$ is G-equivariant and

   $$ \begin{align*} {}_{A|}S<^{\prime}_{|G}{}_{A|}S'&\Leftrightarrow \operatorname{\mathrm{k}} G\otimes _{A|}S<'\operatorname{\mathrm{k}} G\otimes _{A|}S'\\ &\Leftrightarrow S\otimes \operatorname{\mathrm{k}} G<'S'\otimes \operatorname{\mathrm{k}} G\Leftrightarrow S<S'. \end{align*} $$
   Moreover, if $\leq $ is another G-equivariant partial order on $\operatorname {\mathrm {Sim}}(A)$ such that
   $$ \begin{align*} S<'S'\Leftrightarrow _{A|}S<_{A|}S', \end{align*} $$
   then
   $$ \begin{align*} L<L'&\Leftrightarrow \bigoplus_{g\in G}gL<\bigoplus_{g\in G}gL' \Leftrightarrow _{A|}\operatorname{\mathrm{k}} G\otimes L<_{A|}\operatorname{\mathrm{k}} G\otimes L'\\ &\Leftrightarrow \operatorname{\mathrm{k}} G\otimes L<'\operatorname{\mathrm{k}} G\otimes L' \Leftrightarrow L<^{\prime}_G L' \end{align*} $$
   for all $L, L'\in \operatorname {\mathrm {Sim}}(A)$ .

Corollary 4.4. Let $\leq $ be a G-equivariant partial order on $\operatorname {\mathrm {Sim}}(A)$ . Then $\leq $ coincides with $(\leq _G)_{|G}$ .

Let $\leq '$ be a G-stable partial order on $\operatorname {\mathrm {Sim}}(A*G)$ . Then $\leq '$ coincides with $(<^{\prime }_{|G})_{G}$ . Thus, the assignments

$$ \begin{align*} \{G\text{-equivariant partial} & \text{ orders on }\operatorname{\mathrm{Sim}}(A)\}\\ &\rightarrow \{G\text{-stable partial orders on }\operatorname{\mathrm{Sim}}(A*G)\},\\ \leq &\mapsto \leq_G \end{align*} $$

and

$$ \begin{align*} \{G\text{-stable partial orders on }&\operatorname{\mathrm{Sim}}(A*G)\}\\ &\rightarrow \{G\text{-equivariant partial orders on }\operatorname{\mathrm{Sim}}(A)\},\\ \leq' &\mapsto \leq^{\prime}_{|G} \end{align*} $$

are mutually inverse bijections.

From now on, let $\leq _A$ be a G-equivariant partial order on $\operatorname {\mathrm {Sim}}(A)$ and $\leq _{A*G}$ the corresponding induced order on $\operatorname {\mathrm {Sim}}(A*G)$ , or the other way around.

Lemma 4.5. For every $g\in G$ , $L\in \operatorname {\mathrm {Sim}}(A)$ we have $g\Delta _{L}\cong \Delta _{g(L)}$ and $g\widehat {\Delta }_{L}\cong \widehat {\Delta }_{g(L)}$

Proof. This is a direct consequence of the fact that g induces an order-preserving automorphism of $\operatorname {\mathrm {mod}} A$ .

Proposition 4.6. For every simple A-module $L\in \operatorname {\mathrm {Sim}}(A)$ ,

$$ \begin{align*} \operatorname{\mathrm{k}} G\otimes \widehat{\Delta}_{L}\cong\bigoplus_{S\in \operatorname{\mathrm{Sim}}(A*G)}[\operatorname{\mathrm{k}} G\otimes L:S]\widehat{\Delta}_{S}, \end{align*} $$

where $[\operatorname {\mathrm {k}} G\otimes L:S]$ denotes the multiplicity of the simple summand S in $\operatorname {\mathrm {k}} G\otimes L$ .

Proof. Since $\operatorname {\mathrm {k}} G\otimes P_{L}$ is projective with top

$$ \begin{align*}\operatorname{\mathrm{top}}(\operatorname{\mathrm{k}} G\otimes P_{L})\cong \operatorname{\mathrm{k}} G\otimes \operatorname{\mathrm{top}}(P_{L})=\operatorname{\mathrm{k}} G\otimes L\end{align*} $$

by Proposition 2.12(8), there is an isomorphism

$$ \begin{align*} \varphi_L': \bigoplus_{S\in \operatorname{\mathrm{Sim}}(A*G)}[\operatorname{\mathrm{k}} G\otimes L:S]P_{S}\rightarrow \operatorname{\mathrm{k}} G\otimes P_{L}. \end{align*} $$

For the sake of notation, fix a set $\mathcal {S}_L$ of simple $A*G$ -modules such that for any $S\in \operatorname {\mathrm {Sim}}(A*G)$ there are exactly $[\operatorname {\mathrm {k}} G\otimes L:S]$ modules in $\mathcal {S}_L$ which are isomorphic to S and consider instead the isomorphism

$$ \begin{align*} \varphi_L: \bigoplus_{S\in \mathcal{S}_L}P_{S}\rightarrow \operatorname{\mathrm{k}} G\otimes P_{L}. \end{align*} $$

For every $S\in \mathcal {S}_L$ , consider the map

$$ \begin{align*} f_S:=(\operatorname{\mathrm{k}} G\otimes \widehat{\pi}_L)\circ \varphi_L\circ \iota_{P_{S}}:P_{S}\rightarrow \operatorname{\mathrm{k}} G\otimes \widehat{\Delta}_{L}, \end{align*} $$

where $\iota _{P_{S}}:P_{S}\rightarrow \bigoplus _{S'\in \mathcal {S}_L}P_{S'}$ is the canonical embedding.

Then since $\widehat {\Delta }_{L}$ has a filtration by $L'\ngtr _A L$ and $\operatorname {\mathrm {k}} G\otimes -$ is exact, $\operatorname {\mathrm {k}} G\otimes \widehat {\Delta }_{L}$ has a filtration by $\operatorname {\mathrm {k}} G\otimes L'$ such that $L'\ngtr _A L$ . Since for every $L'\ngtr _A L$ any two simple summands $S|\operatorname {\mathrm {k}} G\otimes L$ and $S'|\operatorname {\mathrm {k}} G\otimes L'$ fulfill $S'\ngtr _A S$ , this means that no composition factor $S'$ of $\operatorname {\mathrm {k}} G\otimes \widehat {\Delta }_{L}$ is greater than any summand S of $\operatorname {\mathrm {k}} G\otimes L$ .

Hence, Lemma 3.6 implies that for any $S\in \mathcal {S}_L$ there is a homomorphism $\gamma _S:\widehat {\Delta }_{S}\rightarrow \operatorname {\mathrm {k}} G\otimes \widehat {\Delta }_{L}$ such that $f_S=\gamma _S\circ \widehat {\pi }_S$ . Thus, we obtain a homomorphism

$$ \begin{align*} \gamma:\bigoplus_{S\in \mathcal{S}_L}\widehat{\Delta}_{S}\rightarrow \operatorname{\mathrm{k}} G\otimes \widehat{\Delta}_{L}, (x_S)_S\mapsto \sum_S \gamma_S(x_S) \end{align*} $$

such that

$$ \begin{align*} \gamma\circ (\widehat{\pi}_S)_S((x_S)_S)&=\sum_S \gamma_S\circ \widehat{\pi}_S(x_S) =\sum_S f_S(x_S)\\ &=\sum_S(\operatorname{\mathrm{k}} G\otimes \widehat{\pi}_L)\circ \varphi_L\circ \iota_{P_S}(x_S) =(\operatorname{\mathrm{k}} G\otimes \widehat{\pi}_L)\circ \varphi_L((x_S)_S). \end{align*} $$

In other words, the diagram

commutes. On the other hand, note that by Proposition 2.12(2) and Lemma 4.5 we have a commutative diagram

where the horizontal arrows are isomorphisms. Since $R_G(\widehat {\Delta }_S)$ has a filtration by $R_G(S')$ where $S'\in \operatorname {\mathrm {Sim}}(A*G)$ such that $S'\ngtr _{A*G} S$ and any simple summand $L'$ of $R_G(S')$ fulfills $S'|\operatorname {\mathrm {k}} G\otimes L'$ , we have for any composition factor $L'$ of $R_G(\widehat {\Delta }_S)$ that $\operatorname {\mathrm {k}} G\otimes L'\ngtr S$ . Thus, for $S|\operatorname {\mathrm {k}} G\otimes L$ , $R_G(\widehat {\Delta }_S)$ has no composition factor $L'$ such that $\operatorname {\mathrm {k}} G \otimes L'$ is greater than $\operatorname {\mathrm {k}} G\otimes L$ and thus no composition factor $L'$ which is greater than L.

Hence, analogously to the construction of $\gamma $ , we can construct an A-linear map $\gamma ':R_G(\operatorname {\mathrm {k}} G\otimes \widehat {\Delta }_{L})\rightarrow R_G(\bigoplus _S \widehat {\Delta }_S)$ such that the diagram

commutes.

We obtain a diagram

where both squares commute, that is,

$$ \begin{align*} \gamma'\circ R_G(\operatorname{\mathrm{k}} G\otimes \widehat{\pi}_L)&=R_G((\widehat{\pi}_S)_S)\circ R_G(\varphi_L)^{-1}\\ \text{and } R_G(\gamma)\circ R_G((\widehat{\pi}_S)_S)&= R_G(\operatorname{\mathrm{k}} G\otimes \widehat{\pi}_L)\circ R_G(\varphi_L). \end{align*} $$

In particular,

$$ \begin{align*} R_G(\gamma)\circ \gamma'\circ R_G(\operatorname{\mathrm{k}} G\otimes \widehat{\pi}_L)&=R_G(\gamma)\circ R_G((\widehat{\pi}_S)_S)\circ R_G(\varphi_L)^{-1}\\ &= R_G(\operatorname{\mathrm{k}} G\otimes \widehat{\pi}_L)\circ R_G(\varphi_L)\circ R_G(\varphi_L)^{-1}\\ &=R_G(\operatorname{\mathrm{k}} G\otimes \widehat{\pi}_L) \end{align*} $$

and

$$ \begin{align*} \gamma'\circ R_G(\gamma)\circ R_G((\widehat{\pi}_S)_S)&=\gamma'\circ R_G(\operatorname{\mathrm{k}} G\otimes \widehat{\pi}_L)\circ R_G(\varphi_L)\\ &=R_G((\widehat{\pi}_S)_S)\circ R_G(\varphi_L)^{-1}\circ R_G(\varphi_L)\\ &=R_G((\widehat{\pi}_S)_S). \end{align*} $$

Since $R_G(\operatorname {\mathrm {k}} G\otimes \widehat {\pi }_L)$ and $R_G((\widehat {\pi }_S)_S)$ are epimorphisms, this implies that $\gamma '=R_G(\gamma )^{-1}$ . In particular, $\gamma $ is bijective and hence an isomorphism.

Thus,

$$ \begin{align*} \operatorname{\mathrm{k}} G\otimes \widehat{\Delta}_L\cong \bigoplus_{S\in \mathcal{S}_L}\widehat{\Delta}_S\cong \bigoplus_{S\in \operatorname{\mathrm{Sim}}(A*G)}[\operatorname{\mathrm{k}} G\otimes L:S]\widehat{\Delta}_S. \\[-53pt]\end{align*} $$

Corollary 4.7. For every simple $A*G$ -module $S\in \operatorname {\mathrm {Sim}}(A*G)$ ,

$$ \begin{align*} {}_{A|}\widehat{\Delta}_{S}\cong\bigoplus_{L\in \operatorname{\mathrm{Sim}}(A)}[_{A|}S:L]\widehat{\Delta}_{L}, \end{align*} $$

where $[_{A|}S:L]$ denotes the multiplicity of the simple summand L in ${}_{A|}S$ .

Proof. By Corollary 2.14, there is a simple A-module L such that $[\operatorname {\mathrm {k}} G\otimes L:S]\neq 0$ . By Proposition 4.6, this implies that $\widehat {\Delta }_S$ is a direct summand of $\operatorname {\mathrm {k}} G\otimes \widehat {\Delta }_L$ . Hence, $ {}_{A|}\widehat {\Delta }_{S}$ is a direct summand of

$$ \begin{align*} R_G(\operatorname{\mathrm{k}} G\otimes \widehat{\Delta}_L)\cong \bigoplus_{g\in G}g\widehat{\Delta}_L\cong \bigoplus_{g\in G}\widehat{\Delta}_{gL}, \end{align*} $$

where the last isomorphism follows from Lemma 4.5. In particular, $ {}_{A|}\widehat {\Delta }_{S}$ is a direct sum of pseudostandard modules. Moreover, by Proposition 2.12(8),

$$ \begin{align*} \operatorname{\mathrm{top}}({}_{A|}\widehat{\Delta}_{S})\cong {}_{A|}\operatorname{\mathrm{top}}(\widehat{\Delta}_{S})\cong {}_{A|}S \end{align*} $$

so that, since every pseudostandard module $\widehat {\Delta }_L$ has simple top L, we obtain

$$ \begin{align*} {}_{A|}\widehat{\Delta}_{S}\cong\bigoplus_{L\in \operatorname{\mathrm{Sim}}(A)}[_{A|}S:L]\widehat{\Delta}_{L}.\\[-53pt] \end{align*} $$

Corollary 4.8. $\widehat {\Delta }^{A*G}|\operatorname {\mathrm {k}} G\otimes \widehat {\Delta }^A$ and $\widehat {\Delta }^A|R_G(\widehat {\Delta }^{A*G})$ .

Proof. The first claim follows directly from Proposition 4.6 and Corollary 2.14, while the second claim follows from Corollary 4.7 and Corollary 2.14.

Example 4.9. Let Q be the quiver

and $A:=\operatorname {\mathrm {k}} Q/J^2$ , where J denotes the arrow ideal. Consider the partial order $\leq $ on $\operatorname {\mathrm {Sim}}(A)=\{L_1, L_2\}$ given by an antichain, that is, all distinct elements are incomparable. Then the standard modules of A are simple. Moreover, the group $G=\{e, g\}\cong \mathbb {Z}/2\mathbb {Z}$ acts on A via the $\operatorname {\mathrm {k}}$ -linear map defined via $g(e_1)=e_2$ , $g(e_2)=e_1$ , $g(\alpha )=\beta $ , and $g(\beta )=\alpha $ , and the G action preserves the partial order. Now, by [21, Lem. 8], $A*G$ is Morita equivalent to $\operatorname {\mathrm {k}} [x]/(x^2)$ , hence $\Delta _{\operatorname {\mathrm {k}}}=\operatorname {\mathrm {k}} [x]/(x^2)$ is not simple. In particular, neither Proposition 4.6 nor Corollary 4.8 holds for $(\Delta _L)_L$ instead of $(\widehat {\Delta }_L)_L$ .

Using Corollary 4.8 and Proposition 2.16, we also obtain a concrete description of $(\widehat {\Delta }_S^{A*G})_S$ . We denote, as before, by $H_L$ the stabilizer of the isomorphism class of L and, using Lemma 2.15, choose an $H_L$ -equivariant representative L of this class. Moreover, we let $P_L^{A*H_L}$ be a projective cover of L as an $A*H_L$ -module. We endow $\operatorname {\mathrm {Sim}}(A*H_L)$ with the partial order induced by the partial order on $\operatorname {\mathrm {Sim}}(A)$ as in Definition 4.2 and let $\widehat {\Delta }_L^{A*H_L}$ be the pseudostandard module of $A*H_L$ corresponding to L with respect to this partial order. Then, by Corollary 4.8, the restriction ${}_{A|}\widehat {\Delta }_L^{A*H_L}$ is isomorphic to a direct sum of standard modules with top

$$ \begin{align*}\operatorname{\mathrm{top}}({}_{A|}\widehat{\Delta}_L^{A*H_L})\cong {}_{A|}\operatorname{\mathrm{top}}(\widehat{\Delta}_L^{A*H_L})\cong {}_{A|}L,\end{align*} $$

so that ${}_{A|}\widehat {\Delta }_L^{A*H_L}\cong \widehat {\Delta }_L^A$ . Thus, we obtain the following corollary.

Corollary 4.10. $\widehat {\Delta }_L^A$ has the structure of an $A*H_L$ -module.

Lemma 4.11. The pseudostandard modules $\widehat {\Delta }_S^{A*G}$ for $A*G$ are of the form $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L} (\widehat {\Delta }_L\otimes V)$ where V is an irreducible representation of $H_L$ . Moreover, two such modules $ \operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(\widehat {\Delta }_L\otimes V)$ and $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_{L'}}( \widehat {\Delta }_{L'}\otimes W)$ of this form are isomorphic if and only if there is $g\in G$ such that $gL\cong L'$ and $gV\cong W$ .

Proof. By Proposition 4.6 and Corollary 2.13(1),

$$ \begin{align*} \bigoplus_{S\in \operatorname{\mathrm{Sim}}(A*G)}[\operatorname{\mathrm{k}} G\otimes L:S]\widehat{\Delta}_{S}\cong \operatorname{\mathrm{k}} G\otimes \widehat{\Delta}_{L} &\cong \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L} \operatorname{\mathrm{k}} H_L\otimes \widehat{\Delta}_{L}\\ &\cong \operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L} (\widehat{\Delta}_{L}\otimes \operatorname{\mathrm{k}} H_L)\\ &\cong \bigoplus_{V\in \operatorname{\mathrm{Sim}}(\operatorname{\mathrm{k}} H_L)}[\operatorname{\mathrm{k}} H_L:V]\operatorname{\mathrm{k}} G\otimes_{\operatorname{\mathrm{k}} H_L}(\widehat{\Delta}_L\otimes V). \end{align*} $$

Moreover, $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L} (\widehat {\Delta }_L\otimes V)$ has simple top $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L} (L\otimes V)$ and is in particular indecomposable.

Thus, it is isomorphic to $\widehat {\Delta }_{\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L} (L\otimes V)}$ .

Moreover, $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L}(\widehat {\Delta }_L\otimes V)\cong \widehat {\Delta }_{\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L} (L\otimes V)}$ and $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_{L'}}( \widehat {\Delta }_{L'}\otimes W)\cong \widehat {\Delta }_{\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_{L'}} (L'\otimes W)}$ are isomorphic if and only if $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_L} (L\otimes V)$ and $\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_{L'}} (L'\otimes W)$ are isomorphic, which, by Proposition 2.16, is the case if and only if there is a $g\in G$ such that $gL\cong L'$ and $gV\cong W$ .

Now we use the description of the pseudostandard modules to compare properties of $(A, \leq _A)$ to $(A*G, \leq _{A*G})$ .

Lemma 4.12. The following statements are equivalent:

1. 1. $\leq _A$ is adapted and $(\Delta ^A_L)$ is exceptional.

2. 2. $\leq _{A*G}$ is adapted and $(\Delta _S^{A*G})$ is exceptional.

Proof. By Lemma 3.13 and Proposition 4.3, $\leq _A$ is adapted and $(\Delta _L^A)_L$ is exceptional if and only if

$$ \begin{align*} \operatorname{\mathrm{Hom}}_A(\widehat{\Delta}_L, \operatorname{\mathrm{rad}}(\widehat{\Delta}_{L'}))\neq (0)\Rightarrow L<_AL' \end{align*} $$

and $\leq _{A*G}$ is adapted and $(\Delta _S^{A*G})_S$ is exceptional if and only if

$$ \begin{align*} &\operatorname{\mathrm{Hom}}_{A*G}(\widehat{\Delta}_S, \operatorname{\mathrm{rad}}(\widehat{\Delta}_{S'}))\neq (0)\\ \Rightarrow &L<_AL'\text{ for some (equivalently all) }L, L' \text{ such that }S|\operatorname{\mathrm{k}} G\otimes L\text{ and } S'|\operatorname{\mathrm{k}} G\otimes L'. \end{align*} $$

Now if $\leq _A$ is adapted and $(\Delta _L^A)_L$ exceptional, then for all $S, S'\in \operatorname {\mathrm {Sim}}(A*G)$ and $L, L'\in \operatorname {\mathrm {Sim}}(A)$ such that $S|\operatorname {\mathrm {k}} G\otimes L, S'|\operatorname {\mathrm {k}} G\otimes L'$ we have

$$ \begin{align*} &\operatorname{\mathrm{Hom}}_{A*G}(\widehat{\Delta}_S, \operatorname{\mathrm{rad}}(\widehat{\Delta}_S')) \subseteq \operatorname{\mathrm{Hom}}_{A*G}(\operatorname{\mathrm{k}} G\otimes \widehat{\Delta}_L, \operatorname{\mathrm{rad}}(\operatorname{\mathrm{k}} G\otimes \widehat{\Delta}_{L'}))\\ \subseteq & \operatorname{\mathrm{Hom}}_{A}(\operatorname{\mathrm{k}} G\otimes \widehat{\Delta}_L, \operatorname{\mathrm{rad}}(\operatorname{\mathrm{k}} G\otimes\widehat{\Delta}_{L'})) \cong \bigoplus_{g, g'\in G}\operatorname{\mathrm{Hom}}_A(\widehat{\Delta}_{g(L)}, \operatorname{\mathrm{rad}}(\widehat{\Delta}_{g'(L')})). \end{align*} $$

Thus, if $\operatorname {\mathrm {Hom}}_{A*G}(\widehat {\Delta }_S, \operatorname {\mathrm {rad}}(\widehat {\Delta }_{S'}))\neq (0)$ , there exist $g, g'\in G$ such that

$$ \begin{align*} \operatorname{\mathrm{Hom}}_A(\widehat{\Delta}_{g(L)}, \operatorname{\mathrm{rad}}(\widehat{\Delta}_{g'(L')}))\neq (0). \end{align*} $$

By assumption, this implies $gL<_A g'L$ , and hence, since $\leq _A$ is G-equivariant, $L<_A L'$ . Thus, $\operatorname {\mathrm {k}} G\otimes L<_{A*G} \operatorname {\mathrm {k}} G\otimes L'$ , so that $S<_{A*G}S'$ .

On the other hand, if $\leq _{A*G}$ is adapted and $(\Delta _S^{A*G})$ exceptional, then

$$ \begin{align*} &\operatorname{\mathrm{Hom}}_A(\widehat{\Delta}_L, \operatorname{\mathrm{rad}}(\widehat{\Delta}_{L'}))\neq (0)\\ \Rightarrow &\operatorname{\mathrm{Hom}}_{A*G}(\operatorname{\mathrm{k}} G\otimes \widehat{\Delta}_L, \operatorname{\mathrm{rad}}(\operatorname{\mathrm{k}} G\otimes \widehat{\Delta}_{L'}) )\neq (0)\\ \Rightarrow &\bigoplus_{S|\operatorname{\mathrm{k}} G\otimes L, S'|\operatorname{\mathrm{k}} G\otimes L'}\operatorname{\mathrm{Hom}}_{A*G}(\widehat{\Delta}_S, \operatorname{\mathrm{rad}}(\widehat{\Delta}_{S'})) \neq (0)\\ \Rightarrow &\exists S|\operatorname{\mathrm{k}} G\otimes L, S'|\operatorname{\mathrm{k}} G\otimes L': \operatorname{\mathrm{Hom}}_{A*G}(\widehat{\Delta}_S, \operatorname{\mathrm{rad}}(\widehat{\Delta}_{S'}))\neq (0)\\ \Rightarrow &\exists S|\operatorname{\mathrm{k}} G\otimes L, S'|\operatorname{\mathrm{k}} G\otimes L':S<_{A*G}S'\\ \Leftrightarrow &L<_AL'.\\[-40pt] \end{align*} $$

Example 4.13. It is not true that $\leq _{A*G}$ is adapted if and only if $\leq _A$ is adapted. Consider, for example, $A:=k[x]/(x^2)$ and $G=\{1, g\}\cong \mathbb {Z}/2\mathbb {Z}$ with $g(x)=-x$ . Then A has a unique simple $\operatorname {\mathrm {k}}$ and the unique order is clearly G-equivariant and adapted. However, $A*G$ is given by the quiver

with $\alpha \beta =0=\beta \alpha $ and $\leq _{A*G}$ being an antichain. Hence, $\Delta _1'=P_1$ has socle $L_2$ and top $L_1$ and $L_2$ is incomparable to $L_1$ .

Theorem 4.14. The following statements hold:

1. 1. $(A, \leq _A)$ is quasi-hereditary if and only if $(A*G, \leq _{A*G})$ is quasi-hereditary.

2. 2. $(A, \leq _A)$ is strongly quasi-hereditary if and only if $(A*G, \leq _{A*G})$ is strongly quasi-hereditary.

3. 3. $(A, \leq _A)$ is directed if and only if $(A*G, \leq _{A*G})$ is directed.

Proof.

1. 1. Suppose $(A, \leq _A)$ is quasi-hereditary. Then $\leq _A$ is adapted and $(\Delta _L^A)$ is exceptional. So, by Lemma 4.12, $\leq _{A*G}$ is adapted and $(\Delta _S^{A*G})$ is exceptional. Moreover, A has a filtration by the $\Delta _L=\widehat {\Delta }_L$ . Hence, $A*G\cong \operatorname {\mathrm {k}} G\otimes A$ has a filtration by $\operatorname {\mathrm {k}} G\otimes \widehat {\Delta }_L$ . Since these decompose into a direct sum of $\widehat {\Delta }_S$ , where $S|\operatorname {\mathrm {k}} G\otimes L$ , by Proposition 4.6 and $\Delta _S=\widehat {\Delta }_S$ by Lemma 3.10, this implies that $A*G$ has a filtration by standard modules $\Delta _S=\widehat {\Delta }_S$ . Hence, $A*G\in F(\Delta ^{A*G})$ .

   On the other hand, suppose $A*G$ is quasi-hereditary. Then $\leq _{A*G}$ is adapted and $(\Delta _S^{A*G})$ is exceptional, so that $\leq _A$ is adapted and $(\Delta _L^A)$ is exceptional. Moreover, $A*G$ has a filtration as an $A*G$ -module by standard modules. Hence, as an A-module $A*G\cong \bigoplus _{g\in G}gA\cong |G|A$ has a filtration by the $R_G(\Delta _S)$ , and by Proposition 2.12(2), Lemma 3.10, and Lemma 4.5,

   $$ \begin{align*} R_G(\Delta_S)=R_G(\widehat{\Delta}_S)|R_G(\operatorname{\mathrm{k}} G\otimes \widehat{\Delta}_{L})\cong \bigoplus_{g\in G}\widehat{\Delta}_{g(L)}=\bigoplus_{g\in G}\Delta_{g(L)} \end{align*} $$
   for $S|\operatorname {\mathrm {k}} G\otimes L'$ , so that, since the standard modules of A are indecomposable, $R_G(\Delta _S)$ is a direct sum of standard modules and hence $|G|A\in \mathcal {F}(\Delta )$ . Since by Proposition 3.15, $\mathcal {F}(\Delta )$ is closed under direct summands, $A\in \mathcal {F}(\Delta )$ .

2. 2. By 1., $(A, \leq _A)$ is quasi-hereditary if and only if $(A*G, \leq _{A*G})$ is quasi-hereditary. Moreover, if the projective dimension of $\Delta ^A$ is less than or equal to 1, so is the projective dimension of $\Delta ^{A*G}=\widehat {\Delta }^{A*G}|\operatorname {\mathrm {k}} G\otimes \widehat {\Delta }^A=\operatorname {\mathrm {k}} G\otimes \Delta ^A$ , and if the projective dimension of $\Delta ^{A*G}$ is less than or equal to 1, so is the projective dimension of $\Delta ^A=\widehat {\Delta }^A|R_G(\widehat {\Delta }^{A*G})=R_G(\Delta ^{A*G})$ .

3. 3. By 1., $(A, \leq _A)$ is quasi-hereditary if and only if $(A*G, \leq _{A*G})$ is quasi-hereditary. Moreover, if $\Delta ^A$ is semisimple, so is $\Delta ^{A*G}=\widehat {\Delta }^{A*G}|\operatorname {\mathrm {k}} G\otimes \widehat {\Delta }^A=\operatorname {\mathrm {k}} G\otimes \Delta ^A$ , and if $\Delta _{A*G}$ is semisimple, so is $\Delta ^A=\widehat {\Delta }^A|R_G(\widehat {\Delta }^{A*G})=R_G(\Delta ^{A*G})$ .

Remark 4.15. Example 4.13 shows that it is not in general true that if $(A, \leq _A)$ is standardly stratified, so is $(A*G, \leq _{A*G})$ .

The following lemma can essentially be found in [21, Lem. 8]. However, we have included a proof, since the assumptions there slightly vary from our assumptions.

Lemma 4.16. Let M be an $A*G$ -module. Then G acts on $\operatorname {\mathrm {End}}_A(R_GM)$ via algebra automorphism as in Remark 2.4, and we have an algebra isomorphism

$$ \begin{align*} \theta_M:\operatorname{\mathrm{End}}_A(R_GM)*G&\rightarrow \operatorname{\mathrm{End}}_{A*G}(\operatorname{\mathrm{k}} G\otimes M),\\ f\otimes g&\mapsto (h\otimes m\mapsto hg^{-1}\otimes f(gm)). \end{align*} $$

Proof. It is easy to see that the action given by Remark 2.4 is an action of algebra automorphisms. Let $f\otimes g\in \operatorname {\mathrm {End}}_A(R_GM)*G$ . Then, for any $h,h'\in G$ , $a\in A$ , $m\in M$ , we have

$$ \begin{align*} (\theta_M(f\otimes g))((a\otimes h')(h\otimes m))&=(\theta_M(f\otimes g))(h'h\otimes (hh')^{-1}(a)m)\\ &=h'hg^{-1}\otimes f(g(hh')^{-1}(a)m)\\ &=h'hg^{-1}\otimes ((hh'g^{-1})^{-1}(a)f(gm))\\ &=(a\otimes h')(hg^{-1}\otimes f(gm))\\ &=(a\otimes h')\theta_M(f\otimes g)(h\otimes m). \end{align*} $$

Thus, $\theta _M(f\otimes g)$ is $A*G$ -linear, so that $\theta _M$ is well defined. Moreover, for any $f, f'\in \operatorname {\mathrm {End}}_A(R_GM)$ , $g,g',h\in G$ , $m\in M$ , we have

$$ \begin{align*} \theta_M(f\otimes g)\circ\theta_M(f'\otimes g')(h\otimes m)&=\theta_M(f\otimes g)(h(g')^{-1}\otimes f'(g'm))\\ &=h(g')^{-1}g^{-1}\otimes f(gf'(g'm))\\ &=h(gg')^{-1}\otimes f\circ (g\cdot f')(gg'm)\\ &=\theta_M(f\circ (g\cdot f')\otimes gg')(h\otimes m)\\ &=\theta_M((f\otimes g)(f'\otimes g'))(h\otimes m). \end{align*} $$

Thus, $\theta _M$ is an algebra homomorphism. Let $t\in \operatorname {\mathrm {End}}_{A*G}((\operatorname {\mathrm {k}} G\otimes M))$ . Then we can identify $R_Gt\in \operatorname {\mathrm {End}}_{A}(\bigoplus _{g\in G}gM)$ with a matrix $R_Gt=(t_{g,h})_{g, h\in G}$ where

$$ \begin{align*} t_{g,h}:gM\rightarrow hM \end{align*} $$

and since t is $A*G$ -linear we have

$$ \begin{align*} t_{g,h}=t_{g'g,g'h} \end{align*} $$

for all $g,g',h\in G$ . For $g\in G$ , let

$$ \begin{align*} f_g(t):M\rightarrow M, f_g(t):=t_{g,e}\circ (\operatorname{\mathrm{tr}}_g^M)^{-1} \end{align*} $$

and set

$$ \begin{align*} \tau_M(t):=\sum_{g\in G}f_g\otimes g. \end{align*} $$

Then, for all $m\in M, h\in G$ , we have

$$ \begin{align*} \theta_M(\tau_M(t))(h\otimes m)&=\sum_{g\in G} \theta_M(f_g(t)\otimes g)(h\otimes m)\\ =\sum_{g\in G}hg^{-1}\otimes f_g(t)(gm)&=\sum_{g\in G}hg^{-1}=\sum_{g\in G} hg^{-1}\otimes t_{g,e}\circ (\operatorname{\mathrm{tr}}_g^M)^{-1}(gm)\\ =\sum_{g\in G} hg^{-1}\otimes t_{g,e}(m)&=\sum_{g\in G} hg^{-1}\otimes t_{h,hg^{-1}}(m)\\ =\sum_{g\in G}g\otimes t_{h,g}(m)&=t(h\otimes m). \end{align*} $$

On the other hand, for $f\otimes g\in \operatorname {\mathrm {End}}_A(R_GM)*G$ , we have

$$ \begin{align*} f_h(\theta_M(f\otimes g))=0 \end{align*} $$

unless $h=g$ and

$$ \begin{align*} f_g(\theta_M(f\otimes g))=f \end{align*} $$

Thus, $\tau _M(\theta _M(f\otimes g))=f\otimes g$ , so that $\tau _M=\theta _M^{-1}$ , and $\theta _M$ is an isomorphism.

Theorem 4.17. Let $(A, \leq _A)$ be quasi-hereditary, and let B be a subalgebra of A such that $g(B)=B$ for all $g\in G$ . Then there is a partial order $\leq _B$ on $\operatorname {\mathrm {Sim}}(B)$ such that $(B, \leq _B)$ is an exact Borel subalgebra of $(A, \leq _A)$ if and only if there is a partial order $\leq _{B*G}$ on $\operatorname {\mathrm {Sim}}(B*G)$ such that $(B*G, \leq _{B*G})$ is an exact Borel subalgebra of $(A*G, \leq _{A*G})$ .

Proof. First of all, note that $B*G$ is a well-defined subalgebra of $A*G$ , since $g(B)\subseteq B$ for all $g\in G$ , and that $A*G$ is quasi-hereditary by Theorem 4.14.

Let $(L_i^B)_{1\leq i\leq n}$ be a set of representatives of the isomorphism classes of simple B-modules, and let $(L_j^{B*G})_{1\leq i\leq m}$ be a set of representatives of the isomorphism classes of simple $B*G$ -modules.

Suppose first that $(B, \leq _B)$ is an exact Borel subalgebra of $(A, \leq _A)$ .

Recall that $R_G$ both preserves and reflects short exact sequences, and note that we have a natural isomorphism

(4.1) $$ \begin{align} (A\otimes_B -)\circ R_G\rightarrow R_G\circ ((A*G)\otimes_{B*G}-) \end{align} $$

given by

$$ \begin{align*} A\otimes_B R_GM\rightarrow R_G((A*G)\otimes_{B*G}M), a\otimes m\mapsto (a\otimes 1_G)\otimes m \end{align*} $$

with inverse given by

$$ \begin{align*} R_G((A*G)\otimes_{B*G}M)\rightarrow A\otimes_B R_GM, (a\otimes g)\otimes m\mapsto a\otimes gm. \end{align*} $$

Thus, if $A\otimes _B-$ is exact, so is $(A*G)\otimes _{B*G}-$ .

By assumption, we have a bijection

$$ \begin{align*} \operatorname{\mathrm{Sim}}(B)\rightarrow \operatorname{\mathrm{Sim}}(A), L_i^B\mapsto L_i^A:=\operatorname{\mathrm{top}}(A\otimes_B L_i^B) \end{align*} $$

such that $L_i^B\leq _BL_j^B$ if and only if $L_i^A\leq _AL_j^A$ . In particular, since

$$ \begin{align*} g(\operatorname{\mathrm{top}}(A\otimes_B L_i^B))=\operatorname{\mathrm{top}}(g(A\otimes_B L_B))\cong \operatorname{\mathrm{top}}(A\otimes_B gL_B)) \end{align*} $$

and $\leq _A$ is G-invariant, so is $\leq _B$ . Thus, it induces a partial order $\leq _{B*G}$ on $\operatorname {\mathrm {Sim}}(B*G)$ according to Proposition 4.3.

Next, we want to show that there is a bijection between isomorphism classes of simple modules given by

$$ \begin{align*} \operatorname{\mathrm{Sim}}(B*G)\rightarrow \operatorname{\mathrm{Sim}}(A*G), L^{B*G}_j\mapsto L^{A*G}_j:=\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}L_j^{B*G}). \end{align*} $$

Note that for this, it suffices to show that for any semisimple $B*G$ -module S, the induced map

$$ \begin{align*} s_2:\operatorname{\mathrm{End}}_{B*G}(S)\rightarrow \operatorname{\mathrm{End}}_{A*G}(\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}S)), f\mapsto \operatorname{\mathrm{top}}(\operatorname{\mathrm{id}}_{A*G}\otimes f) \end{align*} $$

is a bijection. Recall from Proposition 2.5 that $\operatorname {\mathrm {End}}_{B*G}(S)=\operatorname {\mathrm {End}}_B(R_GS)^G$ and $\operatorname {\mathrm {End}}_{A*G}(\operatorname {\mathrm {top}}((A*G)\otimes _{B*G}S))=\operatorname {\mathrm {End}}_{A}(R_G\operatorname {\mathrm {top}}((A*G)\otimes _{B*G}S))^G$ where the G-action is given as in Remark 2.6. Moreover, note that by Proposition 2.12(8) and 4.1, we have a natural isomorphism

$$ \begin{align*} \alpha: R_G\circ \operatorname{\mathrm{top}}\circ ((A*G)\otimes_{B*G}-)\rightarrow \operatorname{\mathrm{top}}\circ (A\otimes_B-)\circ R_G,\\ \alpha_M: R_G\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}M)\rightarrow \operatorname{\mathrm{top}}(A\otimes_B R_GM),\\ (a\otimes g)\otimes m+\operatorname{\mathrm{rad}}(A*G)\otimes M\mapsto a\otimes gm+\operatorname{\mathrm{rad}}(A)\otimes M \end{align*} $$

with inverse given by

$$ \begin{align*} \alpha_M^{-1}(a\otimes m+\operatorname{\mathrm{rad}}(A)\otimes M)=(a\otimes 1_G)\otimes m+\operatorname{\mathrm{rad}}(A*G)\otimes M. \end{align*} $$

Transporting the $A*G$ -module structure of $R_G\operatorname {\mathrm {top}}((A*G)\otimes _{B*G}S)$ along $\alpha _S$ gives rise to an $A*G$ -module structure on $\operatorname {\mathrm {top}}(A\otimes _B R_GS)$ , which is given by

$$ \begin{align*} g( a\otimes s+\operatorname{\mathrm{rad}}(A)\otimes S):=g(a)\otimes gs+\operatorname{\mathrm{rad}}(A)\otimes S. \end{align*} $$

According to Remark 2.4, this induces a G-action on $\operatorname {\mathrm {End}}_A(\operatorname {\mathrm {top}}(A\otimes _B R_GS))$ . With this G-action, the homomorphism

$$ \begin{align*} s_1: \operatorname{\mathrm{End}}_B(R_GS)\rightarrow \operatorname{\mathrm{End}}_A(\operatorname{\mathrm{top}}(A\otimes_B R_GS)),\\ f\mapsto \operatorname{\mathrm{top}}(\operatorname{\mathrm{id}}_A\otimes f) \end{align*} $$

becomes G-equivariant, since we have

$$ \begin{align*} \operatorname{\mathrm{top}}(\operatorname{\mathrm{id}}_A\otimes (g \cdot f))(a\otimes s+\operatorname{\mathrm{rad}}(A)\otimes S)=a\otimes (g\cdot f)(s)+\operatorname{\mathrm{rad}}(A)\otimes S\\ =a\otimes g(f(g^{-1}s))+\operatorname{\mathrm{rad}}(A)\otimes S =g(\operatorname{\mathrm{id}}_A(g^{-1}(a)))\otimes g(f(g^{-1}s))+\operatorname{\mathrm{rad}}(A)\otimes S\\ =g(\operatorname{\mathrm{top}}(\operatorname{\mathrm{id}}_A\otimes f)(g^{-1}(s\otimes a+\operatorname{\mathrm{rad}}(A)\otimes S)))= g\cdot (\operatorname{\mathrm{top}}(\operatorname{\mathrm{id}}_A\otimes f))(a\otimes s+\operatorname{\mathrm{rad}}(A)\otimes S). \end{align*} $$

Additionally, since B is an exact Borel subalgebra of A, $s_1$ is an isomorphism. Hence, it restricts to an isomorphism

$$ \begin{align*} s_1^G:\operatorname{\mathrm{End}}_B(R_GS)^G\rightarrow \operatorname{\mathrm{End}}_A(\operatorname{\mathrm{top}}(A\otimes_B R_GS))^G,\\ f\mapsto \operatorname{\mathrm{top}}(\operatorname{\mathrm{id}}_A\otimes f). \end{align*} $$

Moreover, for any $f\in \operatorname {\mathrm {End}}_B(R_GS)^G$ , we have

$$ \begin{align*} &\alpha_S^{-1} \circ\operatorname{\mathrm{top}}(\operatorname{\mathrm{id}}_A\otimes f)\circ \alpha_S((a\otimes g)\otimes s+\operatorname{\mathrm{rad}}(A*G)\otimes S)\\ =\ &\alpha_S^{-1} \circ\operatorname{\mathrm{top}}(\operatorname{\mathrm{id}}_A\otimes f)(a\otimes g(s)+\operatorname{\mathrm{rad}}(A)\otimes S)\\ =\ &\alpha_S^{-1}(a\otimes f(g(s))+\operatorname{\mathrm{rad}}(A)\otimes S)=\alpha_S^{-1}(a\otimes g(f(s))+\operatorname{\mathrm{rad}}(A)\otimes S)\\ =\ &(a\otimes 1)\otimes g(f(s))+\operatorname{\mathrm{rad}}(A*G)\otimes S=(a\otimes g)\otimes f(s)+\operatorname{\mathrm{rad}}(A*G)\otimes S\\ =\ &\operatorname{\mathrm{top}}(\operatorname{\mathrm{id}}_{A*G}\otimes f)((a\otimes g)\otimes s+\operatorname{\mathrm{rad}}(A*G)\otimes S). \end{align*} $$

Hence, conjugation of $s_1^G$ by $\alpha _S$ gives the desired isomorphism $s_2$ . Thus, we have a bijection

$$ \begin{align*} \operatorname{\mathrm{Sim}}(B*G)\rightarrow \operatorname{\mathrm{Sim}}(A*G)\\ L_j^{B*G}\mapsto L_j^{A*G}=\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}L_j^{B*G}). \end{align*} $$

Moreover, for every $L_i^{B*G}, L_j^{B*G}\in \operatorname {\mathrm {Sim}}(B*G)$ , we have, using Definition 4.2, Proposition 4.3, and the fact that B is an exact Borel subalgebra of A, that

$$ \begin{align*} L_i^{B*G}<_{B*G}L_j^{B*G}\Leftrightarrow R_G L_i^{B*G}<_{B}R_GL_j^{B*G}\Leftrightarrow \operatorname{\mathrm{top}}(A\otimes_B R_GL_i^{B*G})<_A \operatorname{\mathrm{top}}(A\otimes_B R_GL_j^{B*G})\\ \Leftrightarrow R_G(\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}L_i^{B*G}))<_A R_G(\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}L_j^{B*G}))\\ \Leftrightarrow \operatorname{\mathrm{top}}((A*G)\otimes_{B*G}L_i^{B*G})<_{A*G}\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}L_j^{B*G})\Leftrightarrow L_i^{A*G}<L_j^{A*G}. \end{align*} $$

Finally, by Proposition 4.6,

$$ \begin{align*} A*G\otimes_{B*G}(\operatorname{\mathrm{k}} G \otimes L^B)&\cong \operatorname{\mathrm{k}} G\otimes (A\otimes_B L^B)\\ &\cong \operatorname{\mathrm{k}} G\otimes \bigoplus_{L_i^A\in \operatorname{\mathrm{Sim}}(A)}\Delta_{L_i^A}\\ &\cong \bigoplus_{L_i^A\in \operatorname{\mathrm{Sim}}(A)}\bigoplus_{L_j^{A*G}\in \operatorname{\mathrm{Sim}}(A*G)}[\operatorname{\mathrm{k}} G\otimes L_i^A:L_j^{A*G}]\Delta_{L_j^{A*G}}. \end{align*} $$

Since for every simple $B*G$ -module $L_j^{B*G}$ we have that $A*G\otimes _{B*G}L_j^{B*G}$ is a summand of $A*G\otimes _{B*G}\operatorname {\mathrm {k}} G \otimes L^B$ , the module $A*G\otimes _{B*G}L_j^{B*G}$ is isomorphic to a direct sum of standard modules. Since it has indecomposable top $L_j^{A*G}$ , this implies that $A*G\otimes _{B*G}L_j^{B*G}\cong \Delta _{L_j^A*G}$ .

Thus, $(B*G, \leq _{B*G})$ is an exact Borel subalgebra of $(A*G, \leq _{A*G})$ .

On the other hand, suppose that $(B*G, \leq _{B*G})$ is an exact Borel subalgebra of $(A*G, \leq _{A*G})$ . Recall that $I_G$ both preserves and reflects short exact sequences, and note that we have a natural isomorphism

(4.2) $$ \begin{align} ((A*G)\otimes_{B*G} -)\circ I_G\rightarrow I_G\circ (A\otimes_{B}-), \end{align} $$

given by

$$ \begin{align*} (A*G)\otimes_{B*G} \operatorname{\mathrm{k}} G\otimes M\rightarrow \operatorname{\mathrm{k}} G\otimes A\otimes_{B} M, (a\otimes g)\otimes h\otimes m\mapsto gh\otimes gh(a)\otimes m \end{align*} $$

with inverse given by

$$ \begin{align*} \operatorname{\mathrm{k}} G\otimes A\otimes_{B} M \rightarrow (A*G)\otimes_{B*G} \operatorname{\mathrm{k}} G\otimes M, g\otimes a\otimes m\mapsto (g^{-1}(a)\otimes 1)\otimes g\otimes m. \end{align*} $$

Thus, if $(A*G)\otimes _{B*G}-$ is exact, so is $A\otimes _{B}-$ .

By assumption, we have a bijection

$$ \begin{align*} \operatorname{\mathrm{Sim}}(B*G)\rightarrow \operatorname{\mathrm{Sim}}(A*G), L_j^{B*G}\mapsto L_j^{A*G}:=\operatorname{\mathrm{top}}((A*G)\otimes_{B*G} L_j^{B*G}) \end{align*} $$

such that $L_i^{B*G}\leq _{B*G}L_j^{B*G}$ if and only if $L_i^{A*G}\leq _{A*G}L_j^{A*G}$ . In particular, since

$$ \begin{align*} (A*G)\otimes_{B*G}(L_j^{B*G}\otimes V)\cong (((A*G)\otimes_{B*G}L_j^{B*G})\otimes V) \end{align*} $$

for every irreducible representation V of G, and since $\leq _{A*G}$ is G-stable, so is $\leq _{B*G}$ . Thus, it induces a partial order $\leq _B$ on $\operatorname {\mathrm {Sim}}(B)$ according to Proposition 4.3.

Next, we want to show that there is a bijection between isomorphism classes of simple modules given by

$$ \begin{align*} \operatorname{\mathrm{Sim}}(B)\rightarrow \operatorname{\mathrm{Sim}}(A), L^{B}_i\mapsto L^{A}_i:=\operatorname{\mathrm{top}}(A\otimes_{B}L_i^{B}). \end{align*} $$

Note that for this, it suffices to show that for any semisimple B-module S, the induced map

(4.3) $$ \begin{align} s_1:\operatorname{\mathrm{End}}_{B}(S)\rightarrow \operatorname{\mathrm{End}}_{A}(\operatorname{\mathrm{top}}(A\otimes_{B}S)), f\mapsto \operatorname{\mathrm{top}}(\operatorname{\mathrm{id}}_{A}\otimes f) \end{align} $$

is a bijection, and, in fact, this holds as soon as it holds for some semisimple B-module S such that $[S:L_i^B]\neq 0$ for all $L_i^B\in \operatorname {\mathrm {Sim}}(B)$ . By Corollary 2.14, it thus suffices to consider the case $S=R_GL^{B*G}$ , where $L^{B*G}=\bigoplus _{L_j^{B*G}\in \operatorname {\mathrm {Sim}}(B*G)}L_j^{B*G}$ . By Lemma 4.16, we have an isomorphism

$$ \begin{align*} \theta_{L^{B*G}}:\operatorname{\mathrm{End}}_B(R_GL^{B*G})*G\rightarrow \operatorname{\mathrm{End}}_{B*G}(I_GR_GL^{B*G}), f\otimes g\mapsto (h\otimes x\mapsto hg^{-1}\otimes f(gx)) \end{align*} $$

as well as an isomorphism

$$ \begin{align*} \theta_{(A*G)\otimes_{B*G}L^{B*G}}:\operatorname{\mathrm{End}}_B(R_G(A*G)\otimes_{B*G}L^{B*G})*G\rightarrow \operatorname{\mathrm{End}}_{A*G}(I_GR_G(A*G)\otimes_{B*G}L^{B*G}),\\ f\otimes g\mapsto (h\otimes (a\otimes g)\otimes m\mapsto hg^{-1}\otimes f(g\cdot ((a\otimes g)\otimes m))). \end{align*} $$

Moreover, since $B*G$ is a regular exact Borel subalgebra of $A*G$ , we have an isomorphism

$$ \begin{align*} s_2:\operatorname{\mathrm{End}}_{B*G}(I_GR_GL^{B*G})\rightarrow \operatorname{\mathrm{End}}_{A*G}(\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}I_GR_GL^{B*G})) f\mapsto \operatorname{\mathrm{top}}(\operatorname{\mathrm{id}}_{A*G}\otimes f). \end{align*} $$

Additionally, by Proposition 2.12(8) and (4.2), there are natural isomorphisms

$$ \begin{align*} \phi_1:\operatorname{\mathrm{top}}\circ (A\otimes_B-)\circ R_G&\rightarrow R_G\circ \operatorname{\mathrm{top}}\circ ((A*G)\otimes_{B*G}-),\\ \phi_1^M:\operatorname{\mathrm{top}}(A\otimes_B R_GM)&\rightarrow R_G(\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}M)),\\ a\otimes m +\operatorname{\mathrm{rad}}(A)\otimes R_GM&\mapsto (a\otimes 1_G)\otimes m+\operatorname{\mathrm{rad}}(A*G)\otimes M \end{align*} $$

and

$$ \begin{align*} \phi_2:I_G\circ R_G\circ \operatorname{\mathrm{top}}\circ ((A*G)\otimes_{B*G}-)&\rightarrow \operatorname{\mathrm{top}}\circ ((A*G)\otimes_{B*G}-)\circ I_G\circ R_G\\ \phi_2^M:I_GR_G\operatorname{\mathrm{top}}(((A*G)\otimes_{B*G}M))&\rightarrow \operatorname{\mathrm{top}}(A*G)\otimes_{B*G}I_GR_GM,\\ h\otimes ((a\otimes g)\otimes m+I_GR_G\operatorname{\mathrm{rad}}(A*G)&\otimes M)\\ \mapsto & (h(a)\otimes h)\otimes (1_G\otimes g(m))+\operatorname{\mathrm{rad}}(A*G)\otimes I_GR_GM, \end{align*} $$

which give rise to isomorphisms

$$ \begin{align*} \varphi_1:\operatorname{\mathrm{End}}_A(\operatorname{\mathrm{top}}(A\otimes_B R_GL^{B*G}))\rightarrow \operatorname{\mathrm{End}}_A(R_G(\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}L^{B*G}))) \end{align*} $$

and

$$ \begin{align*} \varphi_2:\operatorname{\mathrm{End}}_{A*G}(\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}I_GR_GL^{B*G}))\rightarrow \operatorname{\mathrm{End}}_{A*G}(I_GR_G\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}L^{B*G})). \end{align*} $$

Now consider the diagram

For $f\in \operatorname {\mathrm {End}}_B(R_GL^{B*G})$ , $g, h\in G$ , $a\in A$ , and $x\in L^{B*G}$ , we have

$$ \begin{align*} &\varphi_2\circ s_2\circ \theta_{R_GL^{B*G}}(f\otimes g)(1_G\otimes ((1_A\otimes 1_G)\otimes x+\operatorname{\mathrm{rad}}(A*G)\otimes L^{B*G}))\\ &=(\phi_2^{L^{B*G}})^{-1}\circ (s_2\circ \theta_{R_GL^{B*G}})(f\otimes g)\circ \phi_2^{L^{B*G}}(1_G\otimes ((1_A\otimes 1_G)\otimes x+\operatorname{\mathrm{rad}}(A*G)\otimes L^{B*G}))\\ &=(\phi_2^{L^{B*G}})^{-1}\circ (s_2\circ \theta_{R_GL^{B*G}})(f\otimes g)((1_A\otimes 1_G)\otimes 1_G\otimes x+\operatorname{\mathrm{rad}}(A*G)\otimes I_GR_GL^{B*G})\\ &=(\phi_2^{L^{B*G}})^{-1} \circ (\operatorname{\mathrm{top}}(\operatorname{\mathrm{id}}_{A*G}\otimes \theta_{R_GL^{B*G}}(f\otimes g)))((1_A\otimes 1_G)\otimes 1_G\otimes x+\operatorname{\mathrm{rad}}(A*G)\otimes I_GR_GL^{B*G})\\ &=(\phi_2^{L^{B*G}})^{-1}((1_A\otimes 1_G)\otimes \theta_{R_GL^{B*G}}(f\otimes g)(1_G\otimes x)+\operatorname{\mathrm{rad}}(A*G)\otimes I_GR_GL^{B*G})\\ &=(\phi_2^{L^{B*G}})^{-1}((1_A\otimes 1_G)\otimes g^{-1}\otimes f(gx)+\operatorname{\mathrm{rad}}(A*G)\otimes I_GR_GL^{B*G})\\ &=(\phi_2^{L^{B*G}})^{-1}((g^{-1}(1_A)\otimes g^{-1})\otimes 1_G\otimes f(gx)+\operatorname{\mathrm{rad}}(A*G)\otimes I_GR_GL^{B*G})\\ &=g^{-1}\otimes ((1_A\otimes 1_G)\otimes f(g(x)+\operatorname{\mathrm{rad}}(A*G)\otimes L^{B*G})) \end{align*} $$

and

$$ \begin{align*} &\theta_{\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}L^{B*G})}\circ (\varphi_1\otimes\operatorname{\mathrm{id}}_{\operatorname{\mathrm{k}} G})\circ (s_1\otimes\operatorname{\mathrm{id}}_{\operatorname{\mathrm{k}} G})(f\otimes g)(1_G\otimes (1_A\otimes 1_G)\otimes x\\ &\hspace{3cm} +I_GR_G\operatorname{\mathrm{rad}}(A*G)\otimes L^{B*G})\\ &=\theta_{\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}L^{B*G})}((\varphi_1\circ s_1)(f)\otimes g)(1_G\otimes ((1_A\otimes 1_G)\otimes x+I_GR_G\operatorname{\mathrm{rad}}(A*G)\otimes L^{B*G}))\\ &=g^{-1}\otimes ((\varphi_1\circ s_1)(f))(g((1_A\otimes 1_G)\otimes x+\operatorname{\mathrm{rad}}(A*G)\otimes L^{B*G}))\\ &=g^{-1}\otimes \phi_1^{L^{B*G}}\circ s_1(f)\circ (\phi_1^{L^{B*G}})^{-1}(g((1_A\otimes 1_G)\otimes x+\operatorname{\mathrm{rad}}(A*G)\otimes L^{B*G}))\\ &=g^{-1}\otimes \phi_1^{L^{B*G}}\circ s_1(f)(1_A\otimes gx+\operatorname{\mathrm{rad}}(A)\otimes R_GL^{B*G})\\ &=g^{-1}\otimes \phi_1^{L^{B*G}}\circ \operatorname{\mathrm{top}}(\operatorname{\mathrm{id}}_A\otimes f)(1_A\otimes gx+\operatorname{\mathrm{rad}}(A)\otimes R_GL^{B*G})\\ &=g^{-1}\otimes \phi_1^{L^{B*G}}(1_A\otimes f(gx)+\operatorname{\mathrm{rad}}(A)\otimes R_GL^{B*G})\\ &=g^{-1}\otimes ((1_A\otimes 1_G)\otimes f(gx)+\operatorname{\mathrm{rad}}(A*G)\otimes L^{B*G}). \end{align*} $$

As all maps are $A*G$ -linear, and elements of the form $1_G\otimes ((1_A\otimes 1_G)\otimes x+I_GR_G\operatorname {\mathrm {rad}}(A*G)\otimes L^{B*G})$ generate $I_GR_G\operatorname {\mathrm {top}}((A*G)\otimes _{B*G}L^{B*G})$ as an $A*G$ -module, this proves that the diagram commutes. Since we know all maps except $s_1\otimes \operatorname {\mathrm {id}}_{\operatorname {\mathrm {k}} G}$ to be isomorphisms, we can conclude that $s_1\otimes \operatorname {\mathrm {id}}_{\operatorname {\mathrm {k}} G}$ , and thus $s_1$ is an isomorphism. This shows that we have a bijection between the isomorphism classes of simple modules given by

$$ \begin{align*} \operatorname{\mathrm{Sim}}(B)\rightarrow \operatorname{\mathrm{Sim}}(A), L^{B}_i\mapsto L^{A}_i:=\operatorname{\mathrm{top}}(A\otimes_{B}L_i^{B}). \end{align*} $$

Moreover, using Definition 4.2, Proposition 4.3, and the fact that $B*G$ is an exact Borel subalgebra of $A*G$ , we have that

$$ \begin{align*} L_i^B<_B L_j^B\Leftrightarrow &I_G L_i^B<_{B*G} I_GL_j^B \\ \Leftrightarrow &\operatorname{\mathrm{top}}((A*G)\otimes_{B*G}I_G L_i^B)<_{A*G} \operatorname{\mathrm{top}}((A*G)\otimes_{B*G}L_j^B)\\ \Leftrightarrow &I_G (\operatorname{\mathrm{top}}(A\otimes_B L_i^B))<_{A*G} I_G(\operatorname{\mathrm{top}}(A\otimes_B L_j^B)) \\ \Leftrightarrow &\operatorname{\mathrm{top}}(A\otimes_B L_i^B)<_A \operatorname{\mathrm{top}}(A\otimes_B L_j^B)\Leftrightarrow L_i^A<_A L_j^A. \end{align*} $$

Finally, by Corollary 4.7,

$$ \begin{align*} A\otimes_B R_GL^{B*G}\cong R_G((A*G)\otimes_{B*G}L^{B*G})\cong R_G(\Delta^{A*G})\cong\bigoplus_{i=1}^n [L^{A*G}:L_i^A]\Delta^A_{L_i^A}. \end{align*} $$

Since for every simple B-module $L_i^{B}$ we have that $A\otimes _B L_i^B$ is a summand of $A\otimes _B R_GL^{B*G}$ , the module $A\otimes _B L_i^B$ is isomorphic to a direct sum of standard modules. Since it has indecomposable top $L_i^{A}$ , this implies that $A\otimes _{B}L_i^{B}\cong \Delta _{L_i^A}$ .

Thus, $(B, \leq _{B})$ is an exact Borel subalgebra of $(A, \leq _{A})$ .

Proposition 4.18. Let $(A, \leq _A)$ be quasi-hereditary, and let B be a subalgebra of A such that $g(B)=B$ for all $g\in G$ . Let $(B*G, \leq _{B*G})$ be the corresponding exact Borel subalgebra of $(A*G, \leq _{A*G})$ Then the following statements hold:

1. 1. B is a strong exact Borel subalgebra if and only if $B*G$ is a strong exact Borel subalgebra.

2. 2. B is a normal exact Borel subalgebra if and only if $B*G$ is a normal exact Borel subalgebra.

3. 3. B is a homological exact Borel subalgebra if and only if $B*G$ is a homological exact Borel subalgebra.

4. 4. B is a regular exact Borel subalgebra if and only if $B*G$ is a regular exact Borel subalgebra.

Proof.

1. 1. Suppose B is a strong exact Borel subalgebra. Then, by Lemma 3.19, $A\operatorname {\mathrm {rad}}(B)\subseteq \operatorname {\mathrm {rad}}(A)$ . Hence, $A*G\operatorname {\mathrm {rad}}(B*G)\subseteq \operatorname {\mathrm {rad}}(A*G)$ by Lemma 2.11, so that again by Lemma 3.19 $B*G$ is a strong Borel subalgebra of $A*G$ . On the other hand, suppose that $B*G$ is a strong Borel subalgebra of $A*G$ . Then $A*G\operatorname {\mathrm {rad}}(B*G)\subseteq \operatorname {\mathrm {rad}}(A*G)$ , so that again by Lemma 2.11

   $$ \begin{align*} A\operatorname{\mathrm{rad}}(B)\subseteq (A*G\operatorname{\mathrm{rad}}(B*G))\cap A\subseteq \operatorname{\mathrm{rad}}(A*G)\cap A=\operatorname{\mathrm{rad}}(A). \end{align*} $$

2. 2. Suppose that B is normal. Then the inclusion $\iota :B\rightarrow A$ has a splitting $\pi :A\rightarrow B$ as right B-modules whose kernel is a right ideal of A. Since tensoring over $\operatorname {\mathrm {k}}$ is exact, the inclusion $\iota \otimes \operatorname {\mathrm {id}}_{\operatorname {\mathrm {k}} G}$ has the splitting $\pi \otimes \operatorname {\mathrm {id}}_{\operatorname {\mathrm {k}} G}$ which is a right $B*G$ -module homomorphism whose kernel is a right ideal of $A*G$ .

   On the other hand, suppose that $B*G$ is normal. Then $\iota \otimes \operatorname {\mathrm {id}}_{\operatorname {\mathrm {k}} G}$ has a splitting $\pi ':A*G\rightarrow B*G$ of right $B*G$ -modules whose kernel is a right ideal in $A*G$ . Since the fixed point functor $-^G$ for the G-action given by left multiplication is exact by [21, Lem. 3], $(\pi ')^G$ is a splitting of the embedding $(\iota \otimes \operatorname {\mathrm {id}}_{\operatorname {\mathrm {k}} G})^G$ as right $(B*G)^G$ -modules such that its kernel is a right ideal in $(A*G)^G$ . Now since the upward arrows in the commutative diagram

   
   are isomorphisms of algebras, $(\pi ')^G$ induces a splitting of $\iota $ as right B-modules such that its kernel is a right ideal in A.

3. 3.+4. Assume that B is homological (resp. regular). In the latter case, we have already seen that $B*G$ is normal. Let $P^B$ be a projective resolution of $L^B$ . Then $A\otimes _B P^B$ is a projective resolution of $\Delta ^A$ , $\operatorname {\mathrm {k}} G\otimes P^B$ is a projective resolution of $\operatorname {\mathrm {k}} G\otimes L^B$ , $\operatorname {\mathrm {k}} G\otimes (A\otimes _B P^B)$ is a projective resolution of $\operatorname {\mathrm {k}} G\otimes (A\otimes _B L^B)$ , and similarly for the restriction of the induction. Now G acts on $\operatorname {\mathrm {End}}_B(R_G(\operatorname {\mathrm {k}} G\otimes P^B))$ and on $\operatorname {\mathrm {End}}_A(R_G(\operatorname {\mathrm {k}} G\otimes (A\otimes _B P^B)))$ , and since $R_G(\operatorname {\mathrm {k}} G\otimes L^B)$ is semisimple, the map

   $$ \begin{align*} \operatorname{\mathrm{End}}_B(R_G(\operatorname{\mathrm{k}} G\otimes P^B))\rightarrow \operatorname{\mathrm{End}}_A(A\otimes_B R_G(\operatorname{\mathrm{k}} G\otimes P^B)), f\mapsto \operatorname{\mathrm{id}}_A\otimes f \end{align*} $$
   is an epimorphism in homology of degree 1 and an isomorphism in homology of degree strictly greater than 1 (resp. an isomorphism in homology of degree strictly greater than 0).As in Theorem 4.17 conjugating with the isomorphism (4.1)
   $$ \begin{align*} A\otimes_B R_G(\operatorname{\mathrm{k}} G\otimes P^B)\cong R_G((A*G)\otimes_{B*G}(\operatorname{\mathrm{k}} G\otimes P^B)) \end{align*} $$
   yields an isomorphism
   $$ \begin{align*} \operatorname{\mathrm{End}}_B(R_G(\operatorname{\mathrm{k}} G\otimes P^B))\rightarrow \operatorname{\mathrm{End}}_A(R_G(A*G)\otimes_{B*G}(\operatorname{\mathrm{k}} G\otimes P^B)), f\mapsto \operatorname{\mathrm{id}}_{A*G}\otimes f, \end{align*} $$
   which is, as before, G-equivariant, so that it induces a homomorphism
   $$ \begin{align*} &\operatorname{\mathrm{End}}_{B*G}(\operatorname{\mathrm{k}} G\otimes P^B)=\operatorname{\mathrm{End}}_B(R_G(\operatorname{\mathrm{k}} G\otimes P^B))^G\\ \rightarrow &\operatorname{\mathrm{End}}_{A*G}((A*G)\otimes_{B*G}(\operatorname{\mathrm{k}} G\otimes P^B))=\operatorname{\mathrm{End}}_A(R_G(A*G)\otimes_{B*G}(\operatorname{\mathrm{k}} G\otimes P^B))^G,\\ f&\mapsto \operatorname{\mathrm{id}}_{A*G}\otimes f. \end{align*} $$
   Since the fixed point functor $-^G$ is exact, this is an epimorphism in homology of degree 1 and an isomorphism in homology of degree strictly greater than 1 (resp. an isomorphism in homology of degree strictly greater than 0).Thus, we obtain an epimorphism in degree 1 and an isomorphism in degree strictly greater than 1 (resp. an isomorphism in degree strictly greater than 0)
   $$ \begin{align*} \operatorname{\mathrm{Ext}}_{B*G}^{*}(\operatorname{\mathrm{k}} G\otimes L^B, \operatorname{\mathrm{k}} G\otimes L^B)&\rightarrow \operatorname{\mathrm{Ext}}_{A*G}^{*}((A*G)\otimes_{B*G}L^B, (A*G)\otimes_{B*G}L^B),\\ [f]&\mapsto [\operatorname{\mathrm{id}}_{A*G}\otimes f]. \end{align*} $$
   The result now follows from Corollary 2.14.

   On the other hand, suppose $B*G$ is homological (resp. regular). In the latter case, we have already seen that B is normal. Let $P^{B*G}$ be a projective resolution of $L^{B*G}$ . Then $(A*G)\otimes _{B*G} P^{B*G}$ is a projective resolution of $\Delta _{A*G}$ , $R_GP^{B*G}$ is a projective resolution of $R_G(L^{B*G})$ , and $R_G((A*G)\otimes _{B*G} P^{B*G})\cong A\otimes _B R_G(P^{B*G})$ is a projective resolution of $R_G(A*G)\otimes _{B*G} L^{B*G})\cong A\otimes _B R_G(L^{B*G})$ and similarly for the induction of the restriction. Since $L^{B*G}\otimes \operatorname {\mathrm {k}} G$ is semisimple, we have that by assumption

   $$ \begin{align*} \operatorname{\mathrm{Ext}}^*_{B*G}(L^{B*G}\otimes \operatorname{\mathrm{k}} G, L^{B*G}\otimes \operatorname{\mathrm{k}} G)\rightarrow \operatorname{\mathrm{Ext}}^*_{A*G}(\Delta^{A*G}\otimes \operatorname{\mathrm{k}} G, \Delta^{A*G}\otimes \operatorname{\mathrm{k}} G) \end{align*} $$
   is an epimorphism in degree 1 and an isomorphism in degree strictly greater than 1 (resp. an isomorphism in degree strictly greater than 0).Moreover, G acts on $\operatorname {\mathrm {End}}_{B}(R_G(P^{B*G}))$ and on $\operatorname {\mathrm {End}}_A(R_G((A*G)\otimes _{B*G}P^{B*G}))$ via $g\cdot f=gf(g^{-1}-)$ and, arguing as in Theorem 4.17, we obtain a commutative diagram
   
   where $\theta _{P^{B*G}}$ and $\theta _{P^{B*G}}$ and $\theta _{(A*G)\otimes _{B*G}P^{B*G}}$ are the isomorphisms from Lemma 4.16,
   $$ \begin{align*} s_1:\operatorname{\mathrm{End}}_{B}(R_GP^{B*G})\rightarrow \operatorname{\mathrm{End}}_A(A\otimes_B P^{B*G}), f\mapsto \operatorname{\mathrm{id}}_A\otimes f\\ s_2:\operatorname{\mathrm{End}}_{B*G}(I_GR_GP^{B*G})\rightarrow \operatorname{\mathrm{End}}_{A*G}((A*G)\otimes_{B*G} I_GR_GP^{B*G}), f\mapsto \operatorname{\mathrm{id}}_{A*G}\otimes f \end{align*} $$
   are the maps obtained from the induction functor, and $\varphi _1$ and $\varphi _2$ are the isomorphisms arising from the natural isomorphisms (4.1) and (4.2).

   Note that all maps except $s_1\otimes \operatorname {\mathrm {id}}_{\operatorname {\mathrm {k}} G}$ and $s_2$ are isomorphisms and that by assumption $s_2$ is an epimorphism in homology degree 1 and an isomorphism in homology in degree strictly greater than 1 (resp. an isomorphism in homology in degree strictly greater than 0).Thus, $\operatorname {\mathrm {id}}_{\operatorname {\mathrm {k}} G}\otimes s_1$ is an epimorphism in homology degree 1 and an isomorphism in homology in degree strictly greater than 1 (resp. an isomorphism in homology in degree strictly greater than 0).This implies that $s_1$ is in homology degree 1 and an isomorphism in homology in degree strictly greater than 1 (resp. an isomorphism in homology in degree strictly greater than 0).

   We obtain that the vector space homomorphism

   $$ \begin{align*} H^*(s_1):\operatorname{\mathrm{Ext}}_{B}^*(R_GL^{B*G}, R_GL^{B*G})\rightarrow \operatorname{\mathrm{Ext}}_A^*(A\otimes_B L^{B*G}, A\otimes_B L^{B*G}), [f]\mapsto [\operatorname{\mathrm{id}}_A\otimes f] \end{align*} $$
   is an epimorphism in degree 1 and an isomorphism in degree strictly greater than 1 (resp. an isomorphism in degree strictly greater than 0).

   Moreover, since $A\otimes _B L_i^B\cong \Delta _{L_i^A}$ for every $L_i^B\in \operatorname {\mathrm {Sim}}(B)$ and

   $$ \begin{align*}R_G(L^{B*G})\cong \bigoplus_{L_i^B\in \operatorname{\mathrm{Sim}}(B)}[R_G(L^{B*G}):L_i^B]L_i^B\end{align*} $$
   by Proposition 2.12[7], this induces for every $L_i^B, L_j^B\in \operatorname {\mathrm {Sim}}(B)$ a vector space homomorphism
   $$ \begin{align*} [R_G(L^{B*G}):L_i^B][R_G(L^{B*G}):L_j^B]&\operatorname{\mathrm{Ext}}^*_B(L_i^B, L_j^B)\\\rightarrow &[R_G(L^{B*G}):L_i^B][R_G(L^{B*G}):L_j^B]\operatorname{\mathrm{Ext}}^*_A(\Delta_{L_i^A}, \Delta_{L_j^A}),\\ f\mapsto &\operatorname{\mathrm{id}}_A\otimes f, \end{align*} $$
   which is an epimorphism in degree 1 and an isomorphisms in degree strictly greater than 1 (resp. an isomorphism in degree strictly greater than 0). Since by Corollary 2.14 $[R_G(L^{B*G}):L_i^B]\neq 0$ for all i, this implies that $(B,\leq _B)$ is homological resp. regular.

5 Auslander algebras of Nakayama algebras

In this section, we will give, at some length, an example of the above. However, before that, we will need two more general statements.

Lemma 5.1. Let A be a finite-dimensional algebra, and suppose that G is a commutative group acting on A via automorphisms. Let $[X]$ be the isomorphism class of an indecomposable A-module, and let $H_{[X]}$ be the stabilizer of $[X]$ in G. Then there is a representative $Y\in [X]$ such that Y has an $H_{[X]}$ -action.

Proof. Let $n:=|H_{[X]}|$ . Let X be any representative of $[X]$ and consider the $A*H_{[X]}$ -module $\operatorname {\mathrm {k}} H_{[X]}\otimes X$ . As an A-module, this is isomorphic to the direct sum $nX$ , so that via this isomorphism $nX$ obtains likewise the structure of an $A*H_{[X]}$ -module. Thus, we obtain a group homomorphism $\varphi : H_{[X]}\rightarrow \operatorname {\mathrm {Mat}}_{n}(\operatorname {\mathrm {End}}_A(X))^*$ , where $\operatorname {\mathrm {Mat}}_{n}(\operatorname {\mathrm {End}}_A(X))^*$ denote the invertible elements of $\operatorname {\mathrm {Mat}}_{n}(\operatorname {\mathrm {End}}_A(X))^*$ . Since $\operatorname {\mathrm {End}}_A(X)$ is local with residue field $\operatorname {\mathrm {k}}$ , this induces a group homomorphism $\varphi ':H_{[X]}\rightarrow \operatorname {\mathrm {Gl}}_{n}(\operatorname {\mathrm {k}})\cong \operatorname {\mathrm {Gl}}_n(\operatorname {\mathrm {Aut}}(X))$ . Since G and thus $H_{[X]}$ are commutative, the matrices in the image have a common eigenvector. This corresponds to a summand $Y|nX$ , $Y\cong X$ which is stable under the action by $H_{[X]}$ on $nX$ defined by $\varphi '$ , and thus Y has an $H_{[X]}$ -action.

The following proposition is related to [23, Th. 1.3(c.iii)].

Proposition 5.2. Let G be a group acting via automorphisms on an algebra D of finite representation type. Then there is an induced G-action on an algebra $A'$ Morita equivalent to the Auslander algebra A of D such that $A'*G$ is Morita equivalent to the Auslander algebra of $D*G$ .

Moreover, if G is commutative, there is even an induced action on the Auslander algebra A of D such that $A*G$ is Morita equivalent to the Auslander algebra of $D*G$ .

Proof. Let $\{X\}$ be a set of representatives of the isomorphism classes of indecomposable D-modules, and let $M:=\bigoplus _X X$ and $N:=I_GM=\operatorname {\mathrm {k}} G\otimes M$ . Note that for every indecomposable $A*G$ -module Y, $R_GY$ is a direct summand of M, so that $I_GR_GY$ is a direct summand of $I_GM=N$ . Since $Y|I_GR_GY$ by Proposition 2.12(2), this implies that Y is a summand of N. By definition, $A':=\operatorname {\mathrm {End}}_D(R_GN)$ is Morita equivalent to the Auslander algebra A of D. Moreover, as N is a $D*G$ -module, G acts on $A'=\operatorname {\mathrm {End}}_D(R_GN)^{\operatorname {\mathrm {op}}}$ via conjugation. Now, by Lemma 4.16,

$$ \begin{align*} A'*G=\operatorname{\mathrm{End}}_D(R_G N)^{\operatorname{\mathrm{op}}}*G\cong (\operatorname{\mathrm{End}}_D(R_G N)*G)^{\operatorname{\mathrm{op}}} \cong \operatorname{\mathrm{End}}_{D*G}(N\otimes \operatorname{\mathrm{k}} G)^{\operatorname{\mathrm{op}}}. \end{align*} $$

As every indecomposable $D*G$ -module is a summand of N, the latter is Morita equivalent to the Auslander algebra of $D*G$ .

Now if G is commutative, choose instead a set of representatives $\{Y\}$ for the orbits of the isomorphism classes of indecomposable D-modules under the G-action which are equipped with a $H_{[Y]}$ -action, according to Lemma 5.1.

Then the module $N':=\bigoplus _{Y\in \{Y\}}\operatorname {\mathrm {k}} G\otimes _{\operatorname {\mathrm {k}} H_{[Y]}} Y$ obtains the structure of a $D*G$ -module. Moreover, for any $Y\in \{Y\}$ , let $Z_{[Y]}$ be a set of representatives of $G/H_{[Y]}$ . Let X be an indecomposable D-module. Then, by assumption, there is a $g\in G$ and a $Y\in \{Y\}$ such that $X\cong gY$ . Since $hY\cong Y$ for every $h\in H_{[Y]}$ , we can assume without loss of generality that $g=z\in Z_{[Y]}$ . Since by Corollary 2.13(8) $R_GN'\cong \bigoplus _{Y\in \{Y\}}\bigoplus _{z\in Z_{[Y]}}zY$ , we have that $X|R_GN$ . Thus, $A:=\operatorname {\mathrm {End}}_D(N')$ is isomorphic to the Auslander algebra of D. As before, we can use Lemma 4.16 to see that

$$ \begin{align*} A*G\cong \operatorname{\mathrm{End}}_D(R_GN')^{\operatorname{\mathrm{op}}}*G\cong (\operatorname{\mathrm{End}}_D(R_G N')*G)^{\operatorname{\mathrm{op}}} \cong \operatorname{\mathrm{End}}_{D*G}(I_G(R_G N'))^{\operatorname{\mathrm{op}}}, \end{align*} $$

which is Morita equivalent to the Auslander algebra of $D*G$ , arguing as before that since every indecomposable D-module is isomorphic to a summand of $R_GN'$ , every indecomposable $D*G$ -module is a summand of $I_GR_GN$ .

The example we consider arises as follows. Let $D:=\operatorname {\mathrm {k}}[x]/(x^N)$ and $G=\langle g| g^n\rangle \cong \mathbb {Z}/n\mathbb {Z}$ be a cyclic group with n elements and generator $g\in G$ .

Consider the G-action on D given by $gx=\xi x$ where $\xi $ is a primitive nth root of unity. Then, by [23, pp. 241–244], $D*G$ is a self-injective Nakayama algebra with quiver Q

and relations given by all paths of length N, that is, $D*G=kQ/J^N$ . The idempotent $e_j$ for $1\leq i\leq n$ here is given by

$$ \begin{align*} e_j=\frac{1}{n}\sum_{k=0}^{n-1} \xi^{-kj}\otimes g^k \end{align*} $$

and $\alpha _j$ corresponds to the element

$$ \begin{align*} e_{j+1}xe_j=1/n\sum_{k=0}^{n-1}\xi^{-jk}x\otimes g^k=xe_j=e_{j+1}x. \end{align*} $$

In particular, $ge_j=\xi ^{j}e_j$ and

$$ \begin{align*} \alpha_{j+j'-1}\dots\alpha_j=xe_{j+j'-1}xe_{j+j'-2}\dots xe_{j+1}xe_j=x^{j'}e_j=x^{j'}e_j=\sum_{k=0}^{n-1}\xi^{-jk}x^{j'}\otimes g^k \end{align*} $$

so that

$$ \begin{align*} g\alpha_{j+j'}\dots\alpha_j=g\sum_{k=0}^{n-1}\xi^{-jk}x^{j'}\otimes g^k=\sum_{k=0}^{n-1}\xi^{-jk+j'}x^{j'}\otimes g^{k+1}= \xi^{j+j'}\sum_{k=0}^{n-1}\xi^{-jk}x^{j'}\otimes g^{k}. \end{align*} $$

The indecomposable modules of D are given by $M_i=D/(x^{N-i})$ for $0\leq i\leq N-1$ , and the irreducible maps between them are given by the canonical projections and embeddings

$$ \begin{align*} \pi_{i}:M_{i}\rightarrow M_{i+1}, d+(x^{N-i})\mapsto d+(x^{N-i-1})\text{ for }0\leq i\leq N-2\\ \iota_{i}:M_{i} \rightarrow M_{i-1}, d+(x^{N-i})\mapsto dx+(x^{N-i+1})\text{ for }1\leq i\leq N-1 \end{align*} $$

with relations

$$ \begin{align*} \pi_{i-1}\circ \iota_{i}&=\iota_{i+1}\circ \pi_{i} \text{ for }1< i \leq N-1\\ \pi_{N-2}\circ \iota_{N-1}&=0. \end{align*} $$

So the Auslander algebra A of D has quiver $Q'$

with commutator relations at every middle point $M_1, \dots M_{N-2}$ and a zero relation at  $M_{N-1}$ .

Moreover, $gM_i=M_i$ for all $0\leq i\leq N_1$ , so in Proposition 5.2 we may chose $N':=\sum _{i=0}^{N-1}M_i$ and obtain that $A=\operatorname {\mathrm {End}}_A(N')^{\operatorname {\mathrm {op}}}$ is the Auslander algebra of D, and G acts on it via $g(\operatorname {\mathrm {id}}_{M_i})=\operatorname {\mathrm {id}}_{M_i}$ , $g(\pi _i)=\pi _i$ and $g(\iota _i)=\xi \iota _i$ .

Note that since G acts trivially on the primitive idempotents, $A*G$ is basic, so that it is isomorphic to the Auslander algebra of $D*G$ .

By [23, pp. 241–244], the algebra $A*G$ has Gabriel quiver $Q'$ given by

where the last column is identified with the first column, so that $Q'$ becomes a cylinder, with relations given by commutator relations in every parallelogram

and a zero relation given between neighbouring points in the last row

More concretely, the simples correspond to the $D*G$ modules

$$ \begin{align*}M_{i, j}=P_j^{D*G}/(x^{N-i}P_j^{D*G})=((D*G)e_j)/(x^{N-i}(D*G)e_j)\end{align*} $$

for $1\leq i\leq N$ , $1\leq j\leq n$ , which is given by the D-module $M_i$ together with the multiplication

$$ \begin{align*}g\cdot (x^{j'}e_j+x^{N-i}P_j^{D*G})=g\cdot (\alpha_{j+j'-1}\dots \alpha_j)=\xi^{j+j'} x^{j'}e_j+x^{N-i}P_j^{D*G},\end{align*} $$

and the extensions correspond to the maps

$$ \begin{align*} \iota_{i, j}: M_{i,j}&\rightarrow M_{i-1, j-1}, a+x^{N-i}P_j^{D*G}\mapsto axe_{j-1}+x^{N-i+1}P_{j-1}^{D*G} =a\alpha_{j-1}+x^{N-i+1}P_{j-1}^{D*G},\\ \pi_{i,j}:M_{i,j}&\rightarrow M_{i+1, j}, a+x^{N-i}P_j^{D*G}\mapsto a+x^{N-i-1}P_j^{D*G}. \end{align*} $$

Now consider A. The projective $P_i^A$ at $M_i$ is given by

If we define the partial order $M_i\leq M_j$ if and only if $i\leq j$ , then we obtain that A is quasi-hereditary with standard modules given by

where $P_i^A$ has standard filtration

Moreover, note that the subalgebra B of A given by the quiver

is directed. Additionally, A has a vector space basis consisting of

$$ \begin{align*} \iota_{i-k}\circ\dots\circ \iota_{i}\circ \operatorname{\mathrm{id}}_{M_i}\circ \pi_{i-1}\circ \dots \circ \pi_{i-j}\text{ for }0\leq i \leq N-1; 0\leq j, k \leq i. \end{align*} $$

Thus, as a right B-module,

$$ \begin{align*}A_{|B}=\bigoplus_{i=0}^{N-1}\bigoplus_{k=0}^{i}\iota_{i-k}\circ\dots\circ \iota_i\circ \operatorname{\mathrm{id}}_{M_i} B\cong \bigoplus_{i=0}^{N-1}\bigoplus_{j=i}^{N-1}e_{i}B\end{align*} $$

is projective, and

$$ \begin{align*} A\otimes_B L_{M_i}^B\cong A\operatorname{\mathrm{id}}_{M_i}/(A\operatorname{\mathrm{rad}}(B)\operatorname{\mathrm{id}}_{M_i})\cong P_{M_i}^A/A\pi_i=\Delta_{i}^A. \end{align*} $$

Thus, B is an exact Borel subalgebra of A.

Now, since G acts trivially on the simples of A, $\leq _A$ is automatically G-equivariant. Hence, we obtain by Proposition 4.3 an induced partial order $\leq _{A*G}$ on $A*G$ , given by $M_{i,j}<_{A*G} M_{i',j'}\Leftrightarrow i<i'$ , such that by Theorem 4.14 $(A*G, \leq _{A*G})$ is quasi-hereditary. Moreover, by Theorem 4.17, it has an exact Borel subalgebra given by $B*G$ . Using our explicit description of B, and the fact that G acts trivially on B, we obtain that $B*G$ is the subalgebra of $A*G$ given by the subquiver

of $Q'$ . Moreover, if $\iota _B: B\rightarrow A$ is the canonical embedding, $I\subset A$ is the ideal generated by $\iota _1, \dots , \iota _{N-1}$ and $\pi _I:A\rightarrow A/I$ is the canonical projection, then $\pi _I\circ \iota _B$ is an isomorphism, so that $\iota _B$ admits a splitting with kernel I. Hence, B is normal in A, so that $B*G$ is normal in $A*G$ by Theorem 4.17.

However, note that for $N\geq 3$ the extension of $\Delta _1^A$ and $\Delta _2^A$ given by

has a submodule X

which is an extension of $\Delta _0^A$ and $\Delta _2^A$ . Hence, $\operatorname {\mathrm {Ext}}^1_A(\Delta _0^A, \Delta _2^A)\neq (0) =\operatorname {\mathrm {Ext}}^1_B(L_{M_0}^B, L_{M_2}^B)$ , so that B is not regular. Similarly, we can see that $B*G$ is not regular by considering the $A*G$ -module

which is an extension of $\Delta _{0, 1}^{A*G}$ and $\Delta _{2, 2}^{A*G}$ , while $\operatorname {\mathrm {Ext}}^1_{B*G}(L_{M_{0,1}}^{B*G}, L_{M_{2,2}}^{B*G})=(0)$ . For the case $N=2$ , B is a regular exact Borel subalgebra as seen in [18, Exam. A.1].