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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) .. Problems on Lie algebras .. 212 General results of representation Subrepresentations in semisimple representations.

Representation theory was born in 1896 in the work of the German mathematician F. G. Frobenius. This work was triggered by a letter to Frobenius by R. Dedekind. In this letter Dedekind made the following observation: take the multiplication table of a finite group Gand turn it into a matrix XG by replacing every entry gof this table by a ...

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