Related papers: Lie elements and the matrix-tree theorem

Let $\mathfrak{g}$ be a finite dimensional complex Lie algebra and let $A$ be a finite dimensional complex, associative and commutative algebra with unit. We describe the structure of the derivation Lie algebra of the current Lie algebra…

Let $G$ be a complex reductive algebraic group, $g$ its Lie algebra and $h$ a reductive subalgebra of $g$, $n$ a positive integer. Consider the diagonal actions $G:g^n, N_G(h):h^n$. We study a relation between the algebra $C[h^n]^{N_G(h)}$…

An algebra $L$ over a field $\Bbb F$, in which product is denoted by $[\,,\,]$, is said to be \textit{ Lie type algebra} if for all elements $a,b,c\in L$ there exist $\alpha, \beta\in \Bbb F$ such that $\alpha\neq 0$ and $[[a,b],c]=\alpha…

We investigate the finite-dimensional Lie groups whose points are separated by the continuous homomorphisms into groups of invertible elements of locally convex algebras with continuous inversion that satisfy an appropriate completeness…

Let G be a group, F a field of prime characteristic p and V a finite-dimensional FG-module. Let L(V) denote the free Lie algebra on V regarded as an FG-submodule of the free associative algebra (or tensor algebra) T(V). For each positive…

This note provides a formula for the character of the Lie algebra of the fundamental group of a surface, viewed as a module over the symplectic group.

We introduce the concept of a triangular representation of a Lie algebra, give a counterpart of Ado's theorem, and discuss $2$-irreducible triangular modules over a nonreductive Lie algebra.

Let $G$ be a connected reductive algebraic group $G$ over an algebraically closed field $k$ of prime characteristic $p$, and $\ggg=\Lie(G)$. In this paper, we study modular representations of the reductive Lie algebra $\ggg$ with…

Let $\Gamma$ be a discrete group of finite virtual cohomological dimension with certain finiteness conditions of the type satisfied by arithmetic groups. We define a representation ring for $\Gamma$, determined on its elements of finite…

By direct calculations of matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation, and using the automorphism supergroup of the gl(1|1) Lie superalgebra, we determine and classify all…

We study the spectra of non-regular semisimple elements in irreducible representations of simple algebraic groups. More precisely, we prove that if G is a simply connected simple linear algebraic group and f is a non-trivial irreducible…

Let L be a restricted Lie algebra over a field of positive characteristic. We survey the known results about the Lie structure of the restricted enveloping algebra u(L) of L. Related results about the structure of the group of units and the…

We initiate the investigation of the projective variety $E(r,g)$ of elementary subalgebras of dimension $r$ of a ($p$-restricted) Lie algebra $g$ for some $r > 0$ and demonstrate that this variety encodes considerable information about the…

We construct representation theory of Lie algebras with filtrations. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.

Let F_o be a non-archimedean locally compact field of residual characteristic not 2. Let G be a classical group over F_o (with no quaternionic algebra involved) which is not of type A_n for n>1. Let b be an element of the Lie algebra g of G…

We construct an abelian representative for the crossed module associated to the string Lie algebra. We show how to apply this construction in order to define quasi-invariant tensors which serve to categorify the infinitesimal braiding on…

An algebra with identities $a(bc)=b(ac),$ $(ab)c=(ac)b$ is called bicommutative. We construct list of identities satisfied by commutator and anti-commutator products in a free bicommutative algebra. We give criterions for elements of a free…

Let G be a p-adic reductive group, and R an algebraically closed field. Let us consider a smooth representation of G on an R-vector space V. Fix an open compact subgroup K of G and a smooth irreducible representation of K on a…

Let G be a finite group acting on a finite dimensional real vector space V. We denote by P(V) the projective space associated to V. In this paper we compute in a very explicit way the rank of the equivariant complex K-theory of V and P(V),…

The Lie algbera of a compact semisimple Lie group G is determined by the degrees of the irreducible representations of G. However, two different groups can have the same representation degrees.