On Segal–Sugawara vectors and Casimir elements for classical Lie algebras

Richardson Elements for Classical Lie Algebras

Parabolic subalgebras of semi-simple Lie algebras decompose as p = m⊕ n where m is a Levi factor and n the corresponding nilradical. By Richardsons theorem [R], there exists an open orbit under the action of the adjoint group P on the nilradical. The elements of this dense orbits are known as Richardson elements. In this paper we describe a normal form for Richardson elements in the classical c...

Casimir Elements for Certain Polynomial Current Lie Algebras

We consider the polynomial current Lie algebra gl(n)[x] corresponding to the general linear Lie algebra gl(n), and its factor-algebra gm by the idealPk m gl(n)xk. We construct two families of algebraically independent generators of the center of the universal enveloping algebra U(gm) by using the quantum determinant and the quantum contraction for the Yangian of level m. 0. Introduction Let g b...

Graphs for Classical Lie Algebras

A nonzero element x of a Lie algebra L over a field F is called extremal if [x, [x,L]] ⊆ Fx. Extremal elements are a well-studied class of elements in simple finite-dimensional Lie algebras of Chevalley type: they are the long root elements. In [CSUW01], Cohen, Steinbach, Ushirobira and Wales have studied Lie algebras generated by extremal elements, in particular those of Chevalley type. The au...

The centralisers of nilpotent elements in the classical Lie algebras

The definition of index goes back to Dixmier [3, 11.1.6]. This notion is important in Representation Theory and also in Invariant Theory. By Rosenlicht’s theorem [12], generic orbits of an arbitrary action of a linear algebraic group on an irreducible algebraic variety are separated by rational invariants; in particular, ind g = tr.degK(g). The index of a reductive algebra equals its rank. Comp...

Classical Lie Algebras