Related papers: Surprising properties of centralisers in classical…

Let g be a complex semisimple Lie algebra and let G' be the Langlands dual group. We give a description of the cohomology algebra of an arbitrary spherical Schubert variety in the loop Grassmannian for G' as a quotient of the form…

In this work large families of naturally graded nilpotent Lie algebras in arbitrary dimension and characteristic sequence (n,q,1), with n odd, satisfying the centralizer property, are given. This condtion constitutes a generalization, for a…

Various questions on Lie ideals of C*-algebras are investigated. They fall roughly under the following topics: relation of Lie ideals to closed two-sided ideals; Lie ideals spanned by special classes of elements such as commutators,…

In this paper, we study the index for several natural classes of non-reductive subalgebras of semisimple Lie algebras. Namely, we look at parabolic subalgebras, centralisers of nilpotent elements, and the normalisers of the centralisers. We…

Let G be a simple algebraic group over an algebraically closed field with Lie algebra g. Then the orbits of nilpotent elements of g under the adjoint action of G have been classified. We describe a simple algorithm for finding a…

Recently, V.Ginzburg introduced and studied in depth the notion of a principal nilpotent pair in a semisimple Lie algebra \g. Our aim is to contribute to the general theory of nilpotent pairs. Roughly speaking, a nilpotent pair (e_1,e_2)…

The behavior of nilpotents can reveal valuable information about the algebra. We give a simple proof of a classic result that a finite ring is commutative if all its nilpotents lie in the center.

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $\ell$ over an algebraically closed field $\Bbbk$ of characteristic zero, and let $(e,h,f)$ be an $\mathfrak{sl}_2$-triple of g. Denote by $\mathfrak{g}^{e}$ the…

Let $G$ be a connected reductive group over an algebraically closed field $\Bbbk$. Under mild restrictions on the characteristic of $\Bbbk$, we show that any $G$-module with a good filtration also has a good filtration as a module for the…

Let L be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We provide necessary and also some sufficient conditions in order for its Poisson center and semi-center to be polynomial algebras over…

We give a presentation of the centralizer algebras for tensor products of spinor representations of quantum groups via generators and relations. In the even-dimensional case, this can be described in terms of non-standard q-deformations of…

Order three elements in the exceptional groups of type G2 are classified up to conjugation over arbitrary fields. Their centralizers are computed, and the associated classification of idempotents in symmetric composition algebras is…

The representation theory of symmetric Lie superalgebras and corresponding spherical functions are studied in relation with the theory of the deformed quantum Calogero-Moser systems. In the special case of symmetric pair g=gl(n,2m),…

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 $C[h^n]^{N_G(h)}$…

We give criteria for finite dimensionality or infinite dimensionality of the polynomial centralizer of the Lie algebra of a linear Lie group, in terms of invariants and relative invariants of the group. In the finite dimensional scenario…

We study some properties on $\mathsf{Lie}$-centroids related to central $\mathsf{Lie}$-derivations, generalized $\mathsf{Lie}$-derivations and almost inner $\mathsf{Lie}$-derivations. We also determine the $\mathsf{Lie}$-centroid of the…

Some algebraic, geometric and geometroalgebraic characteristics of pairs of operators are discussed.

We classify in this paper Poisson structures on modules over semisimple Lie algebras arising from classical r-matrices. We then study their quantizations and the relation to classical invariant theory.

We check that the connected centralisers of nilpotent elements in the orthogonal and symplectic groups have Levi decompositions in even characteristic. This provides a justification for the identification of the isomorphism classes of the…

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.…