Suppose $\mathfrak {g}=\mathfrak {g}_{\bar 0}+\mathfrak {g}_{\bar 1}$ is a finite-dimensional restricted Lie superalgebra over an algebraically closed field $\mathbf {k}$ of characteristic $p>2$. In this article, we propose a conjecture for maximal dimensions of irreducible modules over the universal enveloping algebra $U(\mathfrak {g})$ of $\mathfrak {g}$, as a super generalization of the celebrated first Kac–Weisfeiler conjecture. It is demonstrated that the conjecture holds for all basic classical Lie superalgebras and all completely solvable restricted Lie superalgebras. In this process, we investigate irreducible representations of solvable Lie superalgebras.

Let ${\mathfrak g}$ be a complex simple Lie algebra and ${\mathfrak n}$ the nilradical of a parabolic subalgebra of ${\mathfrak g}$. We consider some properties of the coadjoint representation of ${\mathfrak n}$ and related algebras of invariants. This includes (i) the problem of existence of generic stabilizers, (ii) a description of the Frobenius semiradical of ${\mathfrak n}$ and the Poisson center of the symmetric algebra , (iii) the structure of as -module, and (iv) the description of square integrable (= quasi-reductive) nilradicals. Our main technical tools are the Kostant cascade in the set of positive roots of ${\mathfrak g}$ and the notion of optimization of ${\mathfrak n}$.

For a finite-dimensional Lie algebra $\mathfrak {L}$ over $\mathbb {C}$ with a fixed Levi decomposition $\mathfrak {L} = \mathfrak {g} \ltimes \mathfrak {r}$ , where $\mathfrak {g}$ is semisimple, we investigate $\mathfrak {L}$ -modules which decompose, as $\mathfrak {g}$ -modules, into a direct sum of simple finite-dimensional $\mathfrak {g}$ -modules with finite multiplicities. We call such modules $\mathfrak {g}$ -Harish-Chandra modules. We give a complete classification of simple $\mathfrak {g}$ -Harish-Chandra modules for the Takiff Lie algebra associated to $\mathfrak {g} = \mathfrak {sl}_2$ , and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright’s and Arkhipov’s completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff $\mathfrak {sl}_2$ and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules.

Let K be a field of characteristic zero. In this paper, we study the polynomial identities of representations of Lie algebras, also called weak identities, or identities of pairs. These identities are determined by pairs of the form (A, L) where A is an associative enveloping algebra for the Lie algebra L. Then a weak identity of (A, L) (or an identity for the representation of L associated to A) is an associative polynomial which vanishes when evaluated on elements of L⊆ A. One of the most influential results in the area of PI algebras was the theory developed by Kemer. A crucial role in it was played by the construction of the Grassmann envelope of an associative algebra and the close relation of the identities of the algebra and its Grassmann envelope. Here we consider varieties of pairs. We prove that under some restrictions one can develop a theory similar to that of Kemer's in the study of identities of representations of Lie algebras. As a consequence, we establish that in the case when K is algebraically closed, if a variety of pairs does not contain pairs corresponding to representations of sl2(K), and if the variety is generated by a pair where the associative algebra is PI then it is soluble. As another consequence of the methods used to obtain the above result, and applying ideas from papers by Giambruno and Zaicev, we were able to construct a pair (A, L) such that its PI exponent (if it exists) cannot be an integer. We recall that the PI exponent exists and is an integer whenever A is an associative (a theorem by Giambruno and Zaicev), or a finite-dimensional Lie algebra (Zaicev). Gordienko also proved that the PI exponent exists and is an integer for finite-dimensional representations of Lie algebras.

Let A be a symmetrisable generalised Cartan matrix, and let $\mathfrak {g}(A)$ be the corresponding Kac–Moody algebra. In this paper, we address the following fundamental question on the structure of $\mathfrak {g}(A)$ : given two homogeneous elements $x,y\in \mathfrak {g}(A)$ , when is their bracket $[x,y]$ a nonzero element? As an application of our results, we give a description of the solvable and nilpotent graded subalgebras of $\mathfrak {g}(A)$ .

We give a new and useful approach to study the representations of symmetric Leibniz algebras. Using this approach, we obtain some results on the representations of these algebras.

We prove that the group of automorphisms of the Lie algebra DerK(Qn) of derivations of the field of rational functions Qn = K(x1, . . ., xn) over a field of characteristic zero is canonically isomorphic to the group of automorphisms of the K-algebra Qn.

The paper deals with the classification of Leibniz central extensions of a filiform Lie algebra. We choose a basis with respect to which the multiplication table has a simple form. In low-dimensional cases isomorphism classes of the central extensions are given. In the case of parametric families of orbits, invariant functions (orbit functions) are provided.

Let M be a maximal subalgebra of the Lie algebra L. A subalgebra C of L is said to be a completion for M if C is not contained in M but every proper subalgebra of C that is an ideal of L is contained in M. The set I(M) of all completions of M is called the index complex of M in L. We use this concept to investigate the influence of the maximal subalgebras on the structure of a Lie algebra, in particular, finding new characterizations of solvable and supersolvable Lie algebras.

Let $KG$ be a non-commutative strongly Lie solvable group algebra of a group $G$ over a field $K$ of positive characteristic $p$. In this note we state necessary and sufficient conditions so that the strong Lie derived length of $KG$ assumes its minimal value, namely $\left\lceil {{\log }_{2}}(p\,+\,1) \right\rceil$.

Let G be a finite group, K a field, and V a finite-dimensional K G-module. Write L(V) for the free Lie algebra on V; similarly, let M ( V) be the free metabelian Lie algebra. The action of G extends naturally to these algebras, so they become KG-modules, which are direct sums of finite-dimensional submodules. This paper explores whether indecomposable direct summands of such a KG-module (for some specific choices of G, K and V) must fall into finitely many isomorphism classes. Of course this is not a question unless there exist infinitely many isomorphism classes of indecomposable KG-modules (that is, K has positive characteristic p and the Sylow p-subgroups of G are non-cyclic) and dim V > 1.

The first two results show that the answer is positive for M(V) when K is finite and dim V = 2, but negative when G is the Klein four-group, the characteristic of K is 2, and V is the unique 3-dimnsional submodule of the regular module D. In the third result, G is again the Klein four-group, K is any field of charateristic 2 with more than 2 elements, V is any faithfull module of dimension 2, and B is the unique 3-dimensional quotient of D; the answer is positive for L(V) if and only if it is positive for each of L(B), L(D), and L(V ⊗ V).

Let G be a finite p-group, and let M(G) be the subgroup generated by the non-central conjugacy classes of G of minimal size. We prove that this subgroup has class at most 3. A similar result is noted for nilpotent Lie algebras.

We develop deformation theory for abelian invariant complex structures on a nilmanifold, and prove that in this case the invariance property is preserved by the Kuranishi process. A purely algebraic condition characterizes the deformations leading again to abelian structures, and we prove that such deformations are unobstructed. Various examples illustrate the resulting theory, and the behavior possible in three complex dimensions.

Using compact simple Lie groups and Heisenberg groups, we combine and generalize the constructions of complex structures on Kodaira surfaces and Hopf surfaces. We identify locally complete parameter spaces of deformations of these spaces and analyze the deformation of Kodaira manifolds in details.

Co-Hopfian finitely generated torsion-free nilpotent groups are characterized in this paper in terms of their Lie algebra automorphisms. Many examples of such groups are also constructed.Partially supported by NSF Grant # DMS-0203979.

In this paper we study nilmanifolds which are modeled on a quotient of a free 2-step nilpotent Lie group by a 1-dimensional subgroup. In fact we obtain a very easy criterion to decide whether or not such a nilmanifold admits an Anosov diffeomorphism.

We generalize to a field of characteristic zero certain properties of the invariant functions of the coadjoint representation of solvable Lie algebras with abelian nilradicals, previously obtained over the base field $\mathbb{C}$ of complex numbers. In particular we determine their number and the restricted type of variables on which they depend. We also determine an upper bound on the maximal number of functionally independent invariants for certain families of solvable Lie algebras with arbitrary nilradicals.

Let F be the free metabelian Lie algebra of finite rank m over a field K of characteristic 0. The automorphism group Aut F is considered with respect to a topology called the formal power series topology and it is shown that the group of tame automorphisms (automorphisms induced from the free Lie algebra of rank m) is dense in Aut F for m ≥ 4 but not dense for m = 2 and m = 3. At a more general level, we study the formal power series topology on the semigroup of all endomorphisms of an arbitrary (associative or non-associative) relatively free algebra of finite rank m and investigate certain associated modules of the general linear group GLm(AT).