Related papers: Sandwich elements and the Richardson property

Let $L$ be a restricted Lie algebra over a field of characteristic $p>2$ and denote by $u(L)$ its restricted enveloping algebra. We establish when the symmetric or skew elements of $u(L)$ under the principal involution are Lie metabelian.

Let $G$ be a complex connected reductive algebraic group and let $G_{\mathbb{R}}$ be a real form of $G$. We construct a sequence of functors $L_i\mathcal{R}$ from admissible (resp. finite-length) representations of $G$ to admissible (resp.…

One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra [K]. A Lie pseudoalgebra is a generalization of the…

We consider parabolic subgroups of a general algebraic group over an algebraically closed field $k$ whose Levi part has exactly $t$ factors. By a classical theorem of Richardson, the nilradical of a parabolic subgroup $P$ has an open dense…

The notion of rigidity of Lie algebra is linked to the following problem: when does a Lie brackets $\mu$ on a vector space g satisfy that every Lie bracket $\mu_1$ sufficiently close to $\mu$ is of the form $\mu_1 = P.\mu $ for some P in…

Let $F$ be an algebraically closed field and consider the Lie algebra ${\mathfrak g}=\langle x\rangle\ltimes {\mathfrak a}$, where $\mathrm{ad}\, x$ acts diagonalizably on the abelian Lie algebra ${\mathfrak a}$. Refer to a ${\mathfrak…

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…

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…

Let $G$ be a real reductive Lie group, $L$ a compact subgroup, and $\pi$ an irreducible admissible representation of $G$. In this article we prove a necessary and sufficient condition for the finiteness of the multiplicities of $L$-types…

Recently Brian Hartwig and the second author found a presentation for the three-point $sl_2$ loop algebra by generators and relations. To obtain this presentation they defined a Lie algebra $\boxtimes$ by generators and relations, and…

In Lie theory, a dense orbit in the unipotent radical of a parabolic group under the adjoint action is called a Richardson orbit. We define a quiver-graded version of Richardson orbits generalising the classical definition in the case of…

Let $G$ be a simple classical algebraic group over an algebraically closed field $K$ of characteristic $p \ge 0$ with natural module $W$. Let $H$ be a closed subgroup of $G$ and let $V$ be a non-trivial irreducible tensor-indecomposable…

We announce here a number of results concerning representation theory of the algebra $R=k<x,y>/ (xy-yx-y^2)$, known as Jordan plane (or Jordan algebra). We consider the question on 'classification' of finite-dimensional modules over the…

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible subgroups of exceptional algebraic groups $G$ which are connected, closed and…

Zassenhaus has proved that if U is a subnormal subalgebra of a finite-dimensional Lie algebra L and V is a finite-dimensional irreducible L-module, then all U-module composition factors of V are isomorphic. Schenkman has proved that if U is…

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…

One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra. A Lie pseudoalgebra is a generalization of the notion…

The first part of this work uses the algorithm recently detailed in arXiv:1906.02935 to classify the irreducible weight modules of the minimal model vertex operator algebra $L_k(\mathfrak{sl}_3)$, when the level $k$ is admissible. These are…

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial finite dimensional irreducible rational $KG$-module.…

We classify the simple restricted modules for the minimal $p$-envelope of the non-graded, non-restricted Hamiltonian Lie algebra $H(2; (1,1); \Phi(1))$ over an algebraically closed field $k$ of characteristic $p \geq 5$. We also give the…