It is shown that the classification theorems for semisimple algebraic groups in characteristic zero can be derived quite simply and naturally from the corresponding theorems for Lie algebras by using a little of the theory of tensor categories. This article is extracted from Milne 2007. Introduction The classical approach to classifying the semisimple algebraic groups over C (see Borel 1975, §1...

Motivated by the study of invariant rings of finite groups on the first Weyl algebras A1 ([1]) and finding interesting families of new noetherian rings, a class of algebras similar to U(sl2) were introduced and studied by Smith in [13]. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obta...

A construction of bases for cell modules of the Birman–Murakami–Wenzl (or B–M–W) algebra Bn(q, r) by lifting bases for cell modules of Bn−1(q, r) is given. By iterating this procedure, we produce cellular bases for B–M–W algebras on which a large Abelian subalgebra, generated by elements which generalise the Jucys–Murphy elements from the representation theory of the Iwahori–Hecke algebra of th...

We prove an analogue of the Lp version of Hardy’s theorem on semisimple Lie groups. The theorem says that on a semisimple Lie group, a function and its Fourier transform cannot decay very rapidly on an average.

We prove a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be group, and let T:= Rep(G) the rigid C*-tensor category of continuous unitary representations G. Mod(T) C*-2-category cofinite semisimple finitely decomposable T-module categories. show that G-C*-algebras can identified with equivalence classes 1-morphisms out object T in Mod(T). For X: -> M1, Y: M2,...

For each nonzero $h\in\mathbb{F}\[x]$, where $\mathbb{F}$ is a field, let $\mathsf{A}\_h$ be the unital associative algebra generated by elements $x,y$, satisfying relation $yx-xy=h$. This gives parametric family of subalgebras Weyl $\mathsf{A}\_1$, containing many well-known algebras which have previously been studied independently. In this paper, we give full description Hochschild cohomology...

This work was motivated by the following two problems from the classical representation theory. (Both problems make sense for an arbitrary complex semisimple Lie algebra but since we shall deal only with the Ar case, we formulate them in this generality). 1. Construct a “good” basis in every irreducible finite-dimensional slr+1-module Vλ, which “materializes” the Littlewood-Richardson rule. A p...