The goal of this paper is to study the irreducible representations of semisimple Lie algebras. We will begin by considering two cases of such algebras: sl2(C) and sl3(C). First, we discover the irreducible representations of sl2(C). The process used in doing so will guide us through our development of the irreducible representations of sl3(C). We will note several key similarities in the proces...

W -algebras of finite type are certain finitely generated associative algebras closely related to universal enveloping algebras of semisimple Lie algebras. In this paper we prove a conjecture of Premet that gives an almost complete classification of finite dimensional irreducible modules for W -algebras. Also we get some partial results towards a conjecture by Ginzburg on their finite dimension...

The notion of conformal algebra appears as an algebraical language describing singular part of operator product expansion (OPE) in conformal field theory [BPZ]. Explicit algebraical exposition of this theory leads to the notion of vertex (chiral) algebra (see, e.g., [B], [DL], [FLM]). Roughly speaking, conformal algebras correspond to vertex algebras by the same way as Lie algebras correspond t...

(1.5) Proposition Let R be a semisimple ring. Then R is isomorphic to a finite direct product ∏s i=1 Ri, where each Ri is a simple ring. (1.6) Proposition Let R be a simple ring. Then there exists a division ring D and a positive integer n such that R ∼= Mn(D). (1.7) Definition Let R be a ring with 1. Define the radical of R to be the intersection of all maximal left ideals of R. The above defi...

The Lie algebra of an algebraic group is the (first) linear approximation to the group. The study of Lie algebras is much more elementary than that of algebraic groups. For example, most of the results on Lie algebras that we shall need are proved already in the undergraduate text Erdmann and Wildon 2006. After many preliminaries, in 7 we describe the structure and classification of the semisi...

We study the induction and restriction functor from a Hopf subalgebra of a semisimple Hopf algebra. The image of the induction functor is described when the Hopf subalgebra is normal. In this situation, at the level of characters this image is isomorphic to the image of the restriction functor. A criterion for subcoalgebras to be invariant under the adjoint action is given generalizing Masuoka’...

In this paper, we compute all Gram determinants associated to all cell modules of Birman-Wenzl algebras. As a by-product, we give a necessary and sufficient condition for Birman-Wenzl algebras being semisimple over an arbitrary field.

We give sufficient conditions that allow contractible (resp., reflexive amenable) Banach algebras to be finite-dimensional and semisimple algebras. Moreover, we show that any contractible (resp., reflexive amenable) Banach algebra in which every maximal left ideal has a Banach space complement is indeed a direct sum of finitely many full matrix algebras. Finally, we characterize Hermitian ∗-alg...

We show that actions of finite-dimensional semisimple commutative Hopf algebras H on //-module algebras A are essentially group-gradings. Moreover we show that the centralizer of H in the smash product A # H equals A" ® H. Using these we invoke results about group graded algebras and results about centralizers of separable subalgebras to give connections between the ideal structure of A, A and ...

We construct canonical bases for quantum generalized Kac-Moody algebras using semisimple perverse sheaves.