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...

SUANMALI, SUTHATHIP. On the Relationship between the Class of a Lie Algebra and the Classes of its Subalgebras. (Under the direction of Ernie L. Stitzinger.) A classical nilpotency result considers finite p-groups whose proper subgroups all have class bounded by a fixed number n. We consider the analogous property in nilpotent Lie algebras. In particular, we investigate whether this condition p...

The observations in this talk come from a paper in preparation by A. Čap, M. Cowling, F. De Mari, M. Eastwood and R. McCallum about the Heisenberg group and the flag manifold, and more general papers by Cowling, De Mari, A. Korányi and H.M. Reimann, one published [?] and one in preparation, as well as papers by McCallum (in preparation) and B. Warhurst [?]. A Carnot group N is a connected, simp...

The Heisenberg Lie Group is the most frequently used model for studying representation theory of groups. This group modular-noncompact and its algebra nilpotent. elements can be expressed in form matrices size 3×3. Another specialty also inherited by three-dimensional called algebra. whose Algebra extended to dimension 2n+1 generalized it denoted H h_n. In this study, surjectiveness exponential...

This paper develops a method for constructing nilpo-tent approximations for local representations of invariant systems on matrix Lie groups via a simple operation on the structure constants of the associated Lie algebra. The crucial role such nilpotent approximations play for the problem of feedback nilpotentization is discussed. The presented ideas are illustrated with an example modeling the ...