We present a framework for the computation of the Hopf 2-cocycles involved in the deformations of Nichols algebras over semisimple Hopf algebras. We write down a recurrence formula and investigate the extent of the connection with invariant Hochschild cohomology in terms of exponentials. As an example, we present detailed computations leading to the explicit description of the Hopf 2-cocycles involved in the deformations of a Nichols algebra of Cartan type $A_2$ with $q=-1$ , a.k.a. the positive part of the small quantum group $\mathfrak{u}^+_{\sqrt{-\text{1}}}(\mathfrak{sl}_3)$ . We show that these cocycles are generically pure, that is they are not cohomologous to exponentials of Hochschild 2-cocycles.

We complete the classification of the pointed Hopf algebras with finite Gelfand-Kirillov dimension that are liftings of the Jordan plane over a nilpotent-by-finite group, correcting the statement in [N. Andruskiewitsch, I. Angiono and I. Heckenberger, Liftings of Jordan and super Jordan planes, Proc. Edinb. Math. Soc., II. Ser. 61(3) (2018), 661–672.].

We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular stochastic PDEs (SPDEs) with regularity structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this article is new and gives a variant of the Butcher–Connes–Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and provides a new perspective on them.

Let $G$ be a split semisimple algebraic group over a field and let $A^*$ be an oriented cohomology theory in the Levine–Morel sense. We provide a uniform approach to the $A^*$-motives of geometrically cellular smooth projective $G$-varieties based on the Hopf algebra structure of $A^*(G)$. Using this approach, we provide various applications to the structure of motives of twisted flag varieties.

We settle several fundamental questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. Moreover, we settle these questions in a greater generality: we give a complete classification of the associated universal formality maps. An important new technical ingredient introduced in this paper is a polydifferential endofunctor ${\mathcal {D}}$ in the category of augmented props with the property that for any representation of a prop ${\mathcal {P}}$ in a vector space $V$ the associated prop ${\mathcal {D}}{\mathcal {P}}$ admits an induced representation on the graded commutative algebra $\odot ^\bullet V$ given in terms of polydifferential operators. Applying this functor to the minimal resolution $\widehat {\mathcal {L}\textit{ieb}}_\infty$ of the genus completed prop $\widehat {\mathcal {L}\textit{ieb}}$ of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in one-to-one correspondence with morphisms of dg props

\[F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}}\widehat{\mathcal{L}\textit{ieb}}_\infty \]

$\mathcal {A}\textit{ssb}_\infty$

$\mathfrak{A}$

${\mathcal {L}} ie_\infty$

${\mathcal {L}} ie_\infty$

$\mathsf {Def}({\mathcal {A}} ss{\mathcal {B}}_\infty \rightarrow {\mathcal {E}} nd_{\odot ^\bullet V})$

$\mathsf {Def}({\mathcal {L}} ie{\mathcal {B}}\rightarrow {\mathcal {E}} nd_V)$

$\odot V$

$V$

$\mathsf {Def}(\mathcal {A}\textit{ssb}_\infty \stackrel {F}{\rightarrow } {\mathcal {D}}\widehat {\mathcal {L}\textit{ieb}}_\infty )$

$\{F_\mathfrak{A}\}$

$GRT=GRT_1\rtimes {\mathbb {K}}^*$

$\{\mathfrak{A}\}$

We construct quantum invariants of balanced sutured 3-manifolds with a ${\text {Spin}^c}$ structure out of an involutive (possibly nonunimodular) Hopf superalgebra H. If H is the Borel subalgebra of ${U_q(\mathfrak {gl}(1|1))}$ , we show that our invariant is computed via Fox calculus, and it is a normalization of Reidemeister torsion. The invariant is defined via a modification of a construction of Kuperberg, where we use the ${\text {Spin}^c}$ structure to take care of the nonunimodularity of H or $H^{*}$ .

We classify pointed Hopf algebras with finite Gelfand–Kirillov dimension whose infinitesimal braiding has dimension 2 but is not of diagonal type, or equivalently is a block. These Hopf algebras are new and turn out to be liftings of either a Jordan or a super Jordan plane over a nilpotent-by-finite group.

In this paper we study the cyclic cohomology of certain ×-Hopf algebras: universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici ×-Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic cohomology. Furthermore, we provide a pairing for the cyclic cohomology of ×-Hopf algebras which generalizes the Connes-Moscovici characteristic map to ×-Hopf algebras. This enables us to transfer the ×-Hopf algebra cyclic cocycles to algebra cyclic cocycles.

We give an explicit construction for a $K(A, n)$ simplicial group and explain its topological interpretation.

Quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one.

This paper is concerned with the theory of cup products in the Hopf cyclic cohomology of algebras and coalgebras. We show that the cyclic cohomology of a coalgebra can be obtained from a construction involving the noncommutative Weil algebra. Then we introduce the notion of higher -twisted traces and use a generalization of the Quillen and Crainic constructions (see [14] and [3]) to define the cup product. We discuss the relation of the cup product above and S-operations on cyclic cohomology. We show that the product we define can be realized as a combination of the composition product in bivariant cyclic cohomology and a map from the cyclic cohomology of coalgebras to bivariant cohomology. In the last section, we briefly discuss the relation of our constructions with that in [9]. More precisely, we propose still another construction of such pairings which can be regarded as an intermediate step between the “Crainic” pairing and that of [9]. We show that it coincides with what in [9] and as far its relation to Crainic's construction is concerned, we reduce the question to a discussuion of a certain map in cohomology (see the question at the end of section 5).

The results of the current paper were announced in [12].

The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal structure of the category of endofunctors on any category and by this we retain some of the combinatorial complexity which makes the theory so interesting. As a basic tool we use distributive laws between monads and comonads (entwinings) on : we define a bimonad on as an endofunctor B which is a monad and a comonad with an entwining λ : BB → BB satisfying certain conditions. This λ is also employed to define the category of (mixed) B-bimodules. In the classical situation, an entwining λ is derived from the twist map for vector spaces. Here this need not be the case but there may exist special distributive laws τ : BB → BB satisfying the Yang-Baxter equation (local prebraidings) which induce an entwining λ and lead to an extension of the theory of braided Hopf algebras.

An antipode is defined as a natural transformation S : B → B with special properties. For categories with limits or colimits and bimonads B preserving them, the existence of an antipode is equivalent to B inducing an equivalence between and the category of B-bimodules. This is a general form of the Fundamental Theorem of Hopf algebras.

Finally we observe a nice symmetry: If B is an endofunctor with a right adjoint R, then B is a (Hopf) bimonad if and only if R is a (Hopf) bimonad. Thus a k-vector space H is a Hopf algebra if and only if Homk(H,−) is a Hopf bimonad. This provides a rich source for Hopf monads not defined by tensor products and generalises the well-known fact that a finite dimensional k-vector space H is a Hopf algebra if and only if its dual H* = Homk(H,k) is a Hopf algebra. Moreover, we obtain that any set G is a group if and only if the functor Map(G,−) is a Hopf monad on the category of sets.

Let $A$ be a graded, commutative Hopf algebra. We study an action of the symmetric group $\sSi_n$ on the tensor product of $n-1$ copies of $A$; this action was introduced by the second author in 1 and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory 2.

We show that for a certain class of Hopf algebras the cohomology ring $H^*(\sSi_n;A^{\otimes n-1})$ is independent of the coproduct provided $n$ and $(n-2)!$ are invertible in the ground ring. With the simplest coproduct structure, the group action becomes particularly tractable and we discuss the implications this has for computations.

AMS 2000 Mathematics subject classification: Primary 16W30; 57T05; 20C30; 20J06; 55S25

We construct dihedral and reflexive cohomology theories for *-Hopf algebras. This generalizes the Connes–Moscovici construction of cyclic cohomology for Hopf algebras.

The purpose of this paper is to extend the class of pairs A, H where H is a Hopf algebra over a field and A a right coideal subalgebra for which H is proved to be either projective or flat as an A-module. The projectivity is obtained under the assumption that H is residually finite dimensional, A has semilocal localizations with respect to a central subring, and there exists a Hopf subalgebra B of H such that the antipode of B is bijective and B is a finitely generated A-module. The flatness of H over A is shown to hold when H is a directed union of residually finite dimensional Hopf subalgebras, and there exists a Hopf subalgebra of H whose center contains A. More general projectivity and flatness results are established for (co)equivariant modules over an H-(co)module algebra under similar assumptions.

The natural problem we approach in the present paper is to show how the notion of formally smooth (co)algebra inside monoidal categories can substitute that of (co)separable (co)algebra in the study of splitting bialgebra homomorphisms. This is performed investigating the relation between formal smoothness and separability of certain functors and led to other results related to Hopf algebra theory. Between them we prove that the existence of ad-(co)invariant integrals for a Hopf algebra H is equivalent to the separability of some forgetful functors. In the finite dimensional case, this is also equivalent to the separability of the Drinfeld Double D(H) over H. Hopf algebras which are formally smooth as (co)algebras are characterized. We prove that if π : E → H is a bialgebra surjection with nilpotent kernel such that H is a Hopf algebra which is formally smooth as a K-algebra, then π has a section which is a right H-colinear algebra homomorphism. Moreover, if H is also endowed with an ad-invariant integral, then this section can be chosen to be H-bicolinear. We also deal with the dual case.

We formulate and prove a free quantum analogue of the first fundamental theorems of invariant theory. More precisely, the polynomial function algebras on matrices are replaced by free algebras, while the universal cosovereign Hopf algebras play the role of the general linear group.

AMS 2000 Mathematics subject classification: Primary 16W30

We classify the finite-dimensional quotient Hopf algebras of the deformed algebra of functions of the general linear group (over an algebraically closed field of zero characteristic). This gives an interesting class of Hopf algebras if the deformation parameter is a root of unity (of odd order). We investigate the properties of these Hopf algebras and construct a new counterexample to Kaplansky's tenth conjecture. E-mail: [email protected] 1991 Mathematics Subject Classification: 81R50, 16W30, 17B37.

This article contains examples and applications of the notion of exact sequences of Hopf algebras.

The purpose of this paper is to discuss the existence, structure, and properties of certain projective and injective Hopf algebras in the category of Hopf algebras that support the structure one expects on the homology of an infinite loop space. As an auxiliary project, we show that these projective and injective Hopf algebras can be realized as the homology of infinite loop spaces associated to spectra obtained from Brown-Gitler spectra by Spanier-Whitehead duality and Brown-Comenetz duality, respectively. We concentrate mainly on indecomposable projectives and injectives, and we work only at the prime 2.