Related papers: Elementary invariants for centralizers of nilpoten…

We study the center of $U(\mathfrak{n})$, where $\mathfrak{n}$ is the locally nilpotent radical of a splitting Borel subalgebra of a simple complex Lie algebra $\mathfrak{g}=\mathfrak{sl}_{\infty}(\mathbb{C})$,…

The index of a finite-dimensional Lie algebra $g$ is the minimum of dimensions of stabilisers $g_\alpha$ of elements $\alpha\in g^*$. Let $g$ be a reductive Lie algebra and $z(x)$ a centraliser of a nilpotent element $x\in g$. Elashvili has…

The algebra of GL_n-invariants of d-tuple of n x n nilpotent matrices with respect to the action by simultaneous conjugation is generated by the traces of products of nilpotent generic matrices in the case of an algebraically closed field…

Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We collect some general results on the Poisson center of S(g), including some simple criteria regarding its polynomiality, and also on…

The vertex algebra W_{1+\infty,c} with central charge c may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer n\geq 1, it was conjectured in the physics…

In the classification theorems of Vinberg and Yakimova for commutative nilmanifolds, the relevant nilpotent groups have a very surprising analytic property. The manifolds are of the form $G/K = N \rtimes K/K$ where, in all but three cases,…

Let K be a field, then we exhibit two matrices in the full nxn matrix algebra M_{n}(K) which generate M_{n}(K) as a Lie K-algebra with the commutator Lie product. We also study Lie centralizers of a not necessarily commutative unitary…

We study the quantum finite W-algebras W(gl_N,f), associated to the Lie algebra gl_N, and its arbitrary nilpotent element f. We construct for such an algebra an r_1 x r_1 matrix L(z) of Yangian type, where r_1 is the number of maximal parts…

Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical…

In line with known results on groups, we show the existence of the nilpotent radical in Lie rings with minimal condition on centralizers. We also prove a form of Engel's theorem if the characteristic is zero.

Let g be a Lie algebra over an algebraically closed field of characteristic p>0 and let U(g) be the universal enveloping algebra of g. We prove in this paper that for g=gl_n and g=sl_n the centre of U(g) is a unique factorisation domain and…

Each matrix A in GL_n(Z) naturally defines an automorphism f of the free r-step nilpotent Lie algebra on n generators. We study the relationship between the matrix A and the eigenvalues and rational invariant subspaces for f. We give…

For the standard Drinfeld-Jimbo quantum group ${\rm U}_q(\mathfrak{g})$ associated with a simple Lie algebra $\mathfrak{g}$, we construct explicit generators of the centre $Z({\rm U}_q(\mathfrak{g}))$, and determine the relations satisfied…

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.

In this work, we propose a new method for a unified study of some of the main features of the theory of the center of the enveloping algebra U(gl(n)) and of the algebra of shifted symmetric polynomials, that allows the whole theory to be…

This paper is concerned with the completion of the proof of the Bergman centralizer theorem by using generic matrices based on our previous quantization proof \cite{KBRZh}. Additionally, we establish that the algebra of generic matrices…

Let F_n = <x_1,...,x_n> denote the free group with generators {x_1,...,x_n}. Nielsen and Magnus described generators for the kernel of the canonical epimorphism from the automorphism group of F_n to the general linear group over the…

Working over an infinite field of positive characteristic, an upper bound is given for the nilpotency index of a finitely generated nil algebra of bounded nil index $n$ in terms of the maximal degree in a minimal homogenous generating…

Infinitesimal Cherednik algebras, first introduced in [EGG], are continuous analogues of rational Cherednik algebras, and in the case of gl_n, are deformations of universal enveloping algebras of the Lie algebras sl_{n+1}. Despite these…

An explicit form of the generators of quantum and ordinary semisimple algebras for an arbitrary finite-dimensional representation is found. The generators corresponding to the simple roots are obtained in terms of a solution of a system of…