Let K be any field of characteristic two and let $U_1$ and $W_1$ be the Lie algebras of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$ and of the polynomial ring K[t], respectively. The algebras $U_1$ and $W_1$ are equipped with natural $\mathbb{Z}$ -gradings. In this paper, we provide bases for the graded identities of $U_1$ and $W_1$ , and we prove that they do not admit any finite basis.

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.

We give explicit formulae for differential graded Lie algebra (DGLA) models of $3$ -cells. In particular, for a cube and an $n$ -faceted banana-shaped $3$ -cell with two vertices, $n$ edges each joining those two vertices, and $n$ bi-gon $2$ -cells, we construct a model symmetric under the geometric symmetries of the cell fixing two antipodal vertices. The cube model is to be used in forthcoming work for discrete analogues of differential geometry on cubulated manifolds.

We give a new factorisation of the classical Dynkin operator, an element of the integral group ring of the symmetric group that facilitates projections of tensor powers onto Lie powers. As an application we show that the iterated Lie power ${{L}_{2}}({{L}_{n}})$ is a module direct summand of the Lie power ${{L}_{2n}}$ whenever the characteristic of the ground field does not divide $n$. An explicit projection of the latter onto the former is exhibited in this case.

We prove that the $\mathfrak{S}$-module PreLie is a free Lie algebra in the category of $\mathfrak{S}$-modules and can therefore be written as the composition of the $\mathfrak{S}$-module Lie with a new $\mathfrak{S}$-module $X$. This implies that free pre-Lie algebras in the category of vector spaces, when considered as Lie algebras, are free on generators that can be described using $X$. Furthermore, we define a natural filtration on the $\mathfrak{S}$-module $X$. We also obtain a relationship between $X$ and the $\mathfrak{S}$-module coming from the anticyclic structure of the PreLie operad.

The orthogonal projection of the free associative algebra onto the free Lie algebra is afforded by an idempotent in the rational group algebra of the symmetric group ${{S}_{n}}$, in each homogenous degree $n$. We give various characterizations of this Lie idempotent and show that it is uniquely determined by a certain unit in the group algebra of ${{S}_{n-1}}$. The inverse of this unit, or, equivalently, the Gram matrix of the orthogonal projection, is described explicitly. We also show that the Garsia Lie idempotent is not constant on descent classes (in fact, not even on coplactic classes) in ${{S}_{n}}$.

A coplactic class in the symmetric group ${{\mathcal{S}}_{n}}$ consists of all permutations in ${{\mathcal{S}}_{n}}$ with a given Schensted $Q$-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of ${{\mathcal{S}}_{n}}$ which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the direct sum of all Solomon descent algebras.

Let $G$ be a group and $K$ a field. For any finite-dimensional $KG$-module $V$ and any positive integer $n$, let $L^n(V)$ denote the $n$th homogeneous component of the free Lie $K$-algebra generated by (a basis of) $V$. Then $L^n(V)$ can be considered as a $KG$-module, called the $n$th Lie power of $V$. The paper is concerned with identifying this module up to isomorphism. A simple formula is obtained which expresses $L^n(V)$ in terms of certain linear functions on the Green ring. When $n$ is not divisible by the characteristic of $K$ these linear functions are Adams operations. Some results are also obtained which clarify the relationship between Adams operations defined by means of exterior powers and symmetric powers and operations introduced by Benson. Some of these results are put into a more general setting in an appendix by Stephen Donkin.

Let be a free abelian group of finite rank, let Γ be a sub-semigroup of satisfying certain finiteness conditions, and let be a (Γ × Z2)-graded Lie superalgebra. In this paper, by applying formal differential operators and the Laplacian to the denominator identity of , we derive a new recursive formula for the dimensions of homogeneous subspaces of . When applied to generalized Kac-Moody superalgebras, our formula yields a generalization of Peterson’s root multiplicity formula. We also obtain a Freudenthal-type weight multiplicity formula for highest weight modules over generalized Kac-Moody superalgebras.

Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees.

In group theory Schreier's technique provides a basis for a subgroup of a free group. In this paper an analogue is developed for free Lie algebras. It hinges on the idea of cutting a Hall set into two parts. Using it, we show that proper subalgebras of finite codimension are not finitely generated and, following M. Hall, that a finitely generated subalgebra is a free factor of a subalgebra of finite codimension.

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).