Chapter 2 Basic notions of representation theory 2.1. What is representation theory? In technical terms, representation theory studies representations of associative algebras. Its general content can be very briefly summa- rized as follows. An associative algebra over a field k is a vector space A over k equipped with an associative bilinear multiplication a, b ab, a, b A. We will always consider associative algebras with unit, i.e., with an element 1 such that 1 · a = a · 1 = a for all a A. A basic example of an associative algebra is the algebra EndV of linear operators from a vector space V to itself. Other important examples include algebras defined by generators and relations, such as group algebras and universal enveloping algebras of Lie algebras. A representation of an associative algebra A (also called a left A-module) is a vector space V equipped with a homomorphism ρ : A EndV , i.e., a linear map preserving the multiplication and unit. A subrepresentation of a representation V is a subspace U V which is invariant under all operators ρ(a), a A. Also, if V1,V2 are two representations of A, then the direct sum V1⊕V2 has an obvious structure of a representation of A. 5
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