In the joint paper [10], the authors established a connection between representations of a finite Lie algebra g0 over a finite field Fq and their algebraic extension g = g0 ⊗Fq k for k = F̄q , which particularly describes irreducible representations of g0 through irreducible representations of g. In this process, the so-called Fg-stable modules play a crucial role, which arise from the Frobenius...

We organize the nilpotent orbits in the exceptional complex Lie algebras into series and show that within each series the dimension of the orbit is a linear function of the natural parameter a = 1, 2, 4, 8, respectively for f4, e6, e7, e8. We observe similar regularities for the centralizers of nilpotent elements in a series and graded components in the associated grading of the ambient Lie alg...

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 . One may associate toX certain cocharacters of Gwith favorable properties; this is an essential feature of the classification of geometric nilpotent orbits due to Bala-Carter, Pommerening, and, more recently, Premet. Suppose that the Lie algebra has a non-degenerate...

We explain a general theory of W-algebras in the context supersymmetric vertex algebras. describe structure associated with odd nilpotent elements Lie superalgebras terms their free generating sets. As an application, we produce explicit generators W-algebra principal element superalgebra $$\mathfrak {gl}(n+1|n)$$ .

We consider the Fokker–Planck operator with a strong external magnetic field. show maximal type estimate on this using nilpotent approach vector field polynomial operators and including notion of representation Lie algebra. This makes it possible to give an optimal characterization domain closure considered operator.

Let A be a finitely generated semigroup with 0. An A–module over F1 (also called an A–set), is a pointed set (M, ∗) together with an action of A. We define and study the Hall algebraHA of the category CA of finite A–modules. HA is shown to be the universal enveloping algebra of a Lie algebra nA, called the Hall Lie algebra of CA. In the case of the 〈t〉 the free monoid on one generator 〈t〉, the ...

The purpose of this article is to set foundations for decomposition numbers of perverse sheaves, to give some methods to calculate them in simple cases, and to compute them concretely in two situations: for a simple (Kleinian) surface singularity, and for the closure of the minimal non-trivial orbit in a simple Lie algebra. This work has applications to modular representation theory, for Weyl g...

The Brownian rough path is the canonical lifting of Brownian motion to the free nilpotent Lie group of order 2. Equivalently it is a process taking values in the algebra of Lie polynomials of degree 2, which is described explicitly by the Brownian motion coupled with its area process. The aim of this article is to compute the finite dimensional characteristic functions of the Brownian rough pat...