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
