Related papers: The Decomposition of Lie Powers

Derived brackets provide a mechanism for generating algebraic structures from graded Lie superalgebras, with applications in Poisson geometry, mathematical physics, and the theory of algebroids. In this paper, we present a complete…

We study gradings by noncommutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if $L$ is gradeg by a non-abelian finite group $G$ then the solvable radical $R$ of…

Let $\Lg$ be a simple complex Lie algebra, we denote by $\Lhg$ the corresponding affine Kac--Moody algebra. Let $\Lambda_0$ be the additional fundamental weight of $\Lhg$. For a dominant integral $\Lg$--coweight $\lam^\vee$, the Demazure…

Let $R$ be a ring and let $\mathcal C$ be a small class of right $R$-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let $\mathcal V (\mathcal C)$ denote a set of representatives of isomorphism classes…

We study the decomposition of a generic element $g \in G$ of a connected reductive complex algebraic group $G$ in the form $g = N(g) B(g) \bar{u} N(g)^{-1}$ where $N: G \dashrightarrow \mathcal{N}_-$ and $B : G \dashrightarrow…

By means of a certain module V and its tensor powers in a finite tensor category, we study a question of whether the depth of a Hopf subalgebra R of a finite-dimensional Hopf algebra H is finite. The module V is the counit representation…

Let K be a field and A be a commutative associative K-algebra which is an integral domain. The Lie algebra Der A of all K-derivations of A is an A-module in a natural way and if R is the quotient field of A, then RDer A is a vector space…

Given a faithful finite-dimensional representation $V$ of a finite group $G$ over any field $\mathbb{F}$, we show that any irreducible ${\mathbb{F}}G$-module $W$ appears, as a submodule or a quotient, in $\mathrm{Sym}^m(V)$ for some integer…

Let g be a finite-dimensional complex simple Lie algebra. Fix a non-negative integer l, we consider the set of dominant weights {\lambda} of g such that l{\Lambda}_0+{\lambda} is a dominant weight for the corresponding untwisted affine…

The author has previously shown that solvable Lie A-algebras and complemented solvable Lie algebras decompose as a vector space direct sum of abelian subalgebras, and their ideals relate nicely to this decomposition. However, neither of…

A Lie algebra is said to be generalised reductive if it is a direct sum of a semisimple Lie algebra and a commutative radical. In this paper we extend the BGG category $\mathcal{O}$ over complex semisimple Lie algebras to the category…

Let $\Gamma=\langle T_{k},M_{l}:k\in\mathbb{Z}^{d},l\in B\mathbb{Z}% ^{d}\rangle $ be a group of unitary operators where $T_{k}$ is a translation operator and $M_{l}$ is a modulation operator acting on $L^{2}\left( \mathbb{R}^{d}\right) .$…

We study the representation theory of finite-dimensional $\omega$-Lie algebras over the complex field. We derive an $\omega$-Lie version of the classical Lie's theorem, i.e., any finite-dimensional irreducible module of a soluble…

In the present note we suggest an affinization of a theorem by Kostrikin et.al. about the decomposition of some complex simple Lie algebras ${\cal G}$ into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out that the…

The present article is part of a research program the aim of which is to find all indecomposable solvable extensions of a given class of nilpotent Lie algebras. Specifically in this article we consider a nilpotent Lie algebra n that is…

This paper exhibits fundamental structure underlying Lie algebra homology with coefficients in tensor products of the adjoint representation, mostly focusing upon the case of free Lie algebras. The main result yields a DG category that is…

A weakly complete vector space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ is isomorphic to $\mathbb{K}^X$ for some set $X$ algebraically and topologically. The significance of this type of topological vector spaces is…

This is an old paper put here for archeological purposes. It is proved that a finite-dimensional Lie algebra over a field of characteristic p>5, that can be written as a vector space (not necessarily direct) sum of two nilpotent…

We introduce the class of graded Lie-Rinehart algebras as a natural generalization of the one of graded Lie algebras. For $G$ an abelian group, we show that if $L$ is a tight $G$-graded Lie-Rinehart algebra over an associative and…

Let $G$ be the special linear group of degree $2$ over an algebraically closed field $K$. Let $E$ be the natural module and $S^rE$ the $r$th symmetric power. We consider here, for $r,s\geq 0$, the tensor product of $S^rE$ and the dual of…