In this paper, we outline the rudiments of the representation theory of semisimple Lie algebras. We build the necessary theory in order to analyze the representations of sl2, which includes proving that representations of semisimple Lie algebras are completely reducible and preserve the Jordan decomposition. We only assume the reader has a working knowledge of linear algebra and a little famili...

In the first part of this article we prove that one conditions required in original definition nearly Frobenius algebra, coassociativity, is redundant. Also, determine dimension product and tensor two algebras from each them. We apply these results to semisimple algebras. second introduce notion normalized algebra. a series equivalences: concept algebra equivalent separable fact projective as b...

We study the problem of matrix Lie algebra conjugacy. Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley–Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set L of matrices such that M1,M2 ∈ L =⇒ M1M2 − M2M1 ∈ L. Two matrix Lie algebras are conjugate if there is an invertible matrix M such that L1 = ML...

Kac-Moody algebras are a generalization of the finite-dimensional semisimple Lie algebras that have many characteristics similar to the finite-dimensional ones. These possibly infinite-dimensional Lie algebras have found applications everywhere from modular forms to conformal field theory in physics. In this thesis we give two main results of the theory of Kac-Moody algebras. First, we present ...

Mason and Ng have given a generalization to semisimple quasiHopf algebras of Linchenko and Montgomery’s generalization to semisimple Hopf algebras of the classical Frobenius-Schur theorem for group representations. We give a simplified proof, in particular a somewhat conceptual derivation of the appropriate form of the Frobenius-Schur indicator that indicates if and in which of two possible fas...

Introduction For this course you need a very good understanding of linear algebra; a good knowledge of group theory and the representation theory of finite groups will also help. The main sources for these notes are the books [6] and [8]. We give complete proofs of all statements with the exception of the conjugacy of Cartan subgroups, the uniqueness theorem for semisimple Lie algebras, and the...

In this paper we begin a classification of simple and semisimple totally antiflexible algebras (finite-dimensional) over splitting fields of char. ^2, 3.. For such an algebra A , let P be the largest associative ideal in A and let /V+ be the radical of P. We determine all simple and semisimple totally antiflexible algebras in which N ■ N =0. Defining A to be of type (m, n) if N is nilpotent of ...