Related papers: Elementary invariants for centralizers of nilpoten…
In this thesis we discuss some properties of centralisers in classical Lie algebas and related structures. Our results follow three distinct but related themes: the modular representation theory of centralisers, the sheets of simple Lie…
The existence of invariant generators for distributions satisfying a compatibility condition with the symmetry algebra is proved.
We study the subspace of the exterior algebra of a simple complex Lie algebra linearly spanned by the copies of the little adjoint representation or, in the case of the Lie algebra of traceless matrices, by the copies of the n-th symmetric…
We study the Yangian $Y_n$ associated to the general linear Lie algebra $\mathfrak{gl}_n$ over a field of positive characteristic, as well as its shifted analog $Y_n(\sigma)$. Our main result gives a description of the centre of…
For each simple Lie algebra g consider the corresponding affine vertex algebra V_{crit}(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem…
We show that the central representation is nontrivial for all one-dimensional central extensions of nilpotent Lie algebras possessing a codimension one abelian ideal.
In this article we prove that there exists a Dixmier map for nilpotent super Lie algebras. In other words, if we denote by Prim(U(g)) the set of (graded) primitive ideals of the enveloping algebra U(g) of g and Ad_0 the adjoint group of…
It is well known that any polycyclic group, and hence any finitely generated nilpotent group, can be embedded into $GL_{n}(\mathbb{Z})$ for an appropriate $n\in \mathbb{N}$; that is, each element in the group has a unique matrix…
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…
Over a field of characteristic 0, the algebra of invariants of several $n\times n$ matrices under simultaneous conjugation by $GL_n$ is generated by traces of products of generic matrices. In this paper we have found, in terms of…
We present a criterion that serves as the basis for a polynomial-time algorithm to decide whether a finite set of qudit gates exponentiated by some Hamiltonians is universal. Our approach formulates universality in Lie algebraic terms and…
In this note we describe the structure of finite dimensional Malcev algebras over a field of real numbers $\mathbb{R}$, which are nilpotent module its Lie center. It is proved that the corresponding analytic global Moufang loops are…
A famous result of Kostant states that the universal enveloping algebra of a semisimple complex Lie algebra is a free module over its center. We prove an analogue of this result for a class of filtered algebras and apply it to show the…
We give a detailed description of infinite locally nilpotent groups G such that the index |C_G (x) : <x>| is finite, for every non-normal cyclic subgroup <x> of G. We are also able to extend our analysis to all non-periodic groups…
We prove Bergman's theorem on centralizers by using generic matrices and Kontsevich's quantization method. For any field $\textbf{k} $ of positive characteristics, set $A=\textbf{k} \langle x_1,\dots,x_s\rangle$ be a free associative…
In this paper, we study the concept of free pre-Lie algebra generated by a (non-empty) set. We review the construction of A. Agrachev and R. Gamkrelidze of monomial bases in free pre-Lie algebras. We describe the matrix of the monomial…
The index of a complex Lie algebra is the minimal codimension of its coadjoint orbits. Let us suppose $\g$ semisimple, then its index, ${\rm ind} \g$, is equal to its rank, ${\rm rk \g}$. The goal of this paper is to establish a simple…
We study representations of the central extension of the Lie algebra of differential operators on the circle, the W-infinity algebra. We obtain complete and specialized character formulas for a large class of representations, which we call…
We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra $A$ generated by an irreducible representation of such a group has…
Let $\ggg=\ggg_\bz+\ggg_\bo$ be a basic classical Lie superalgebra over an algebraically closed field $k$ of characteristic $p>2$, and $G$ be an algebraic supergroup satisfying $\Lie(G)=\ggg$, with the purely even subgroup $G_\ev$ which is…