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Introduction to representation theory

Introduction to representation theoryPavel Etingof, Oleg Golberg, Sebastian Hensel,Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena YudovinaJanuary 10, 2011 Contents1 Basic notions of representation What is representation theory ? .. Algebras .. Representations .. Ideals .. Quotients .. Algebras defined by generators and relations .. Examples of algebras .. Quivers .. Lie algebras .. Tensor products .. The tensor algebra .. Hilbert s third problem .. Tensor products and duals of representations of Lie algebras .. Representations ofsl(2).

group algebra A= C[G] of a finite group G– the algebra with basis ag,g∈ Gand multiplication law agah = agh. We will show that any finite dimensional representation of Ais a direct sum of irreducible representations, i.e., the notions of an irreducible and indecomposable representation are the same for A(Maschke’s theorem).

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