Let g be a complex simple Lie algebra, with fixed Borel subalgebra b⊂g and Weyl group W. Expanding on previous work of Fan Stembridge in the simply laced case, this note aims to study fully commutative elements W, their connections spherical nilpotent orbits g. If is not type G2, it shown that an element w∈W if only b determined by inversions w lies closure orbit. A similar characterization als...

We show that the sum of two adjoint orbits in the Lie algebra of an exponential Lie group coincides with the Campbell-Baker-Hausdorff product of these two orbits. Introduction N. Wildberger and others have recently investigated the structure of the hypergroup of the adjoint orbits in relation with the class hypergroup of compact Lie groups. A generalization of the notion of this type of hypergr...

in this article, we introduce monomial irreducible representations of the special linear lie algebra $sln$. we will show that this kind of representations have bases for which the action of the chevalley generators of the lie algebra on the basis elements can be given by a simple formula.

A finite W -algebra U(g, e) is a certain finitely generated algebra that can be viewed as the enveloping algebra of the Slodowy slice to the adjoint orbit of a nilpotent element e of a complex reductive Lie algebra g. It is possible to give the tensor product of a U(g, e)-module with a finite dimensional U(g)module the structure of a U(g, e)-module; we refer to such tensor products as translati...

Let G be the adjoint group of the simple real Lie algebra g , and let K C → Aut(p C ) be the complexified isotropy representation at the identity coset of the corresponding symmetric space. We classify the spherical nilpotent K C orbits in p C .

We describe two algorithms for finding representatives of the nilpotent orbits of a θ-group. The algorithms have been implemented in the computer algebra system GAP (inside the package SLA). We comment on their performance, and we apply the algorithms to obtain lists of N-regular automorphisms of simple Lie algebras of exceptional type.

We generalize the notion of trace to the Kontsevich quantization algebra and show that for all Poisson manifolds representable by quotients of a symplectic manifold by a Hamiltonian action of a nilpotent Lie group, the trace is given by integration with respect to a unimodular volume form.

In a paper published by Ranga Rao [4] in the Annals of Mathematics, it is shown that the invariant measure carried by a nilpotent orbital can be extended to an invariant distribution on the ambient Lie algebra. This paper provides an alternative proof by a method proposed by Gordon Savin.

We introduce the notion of Gröbner S-basis of an ideal of the free associative algebra K〈X〉 over a field K invariant under the action of a semigroup S of endomorphisms of the algebra. We calculate the Gröbner S-bases of the ideal corresponding to the universal enveloping algebra of the free nilpotent of class 2 Lie algebra and of the T-ideal generated by the polynomial identity [x, y, z] = 0, w...