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.

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http://dx.doi.org/10.1090/stml/059/02