Representation Theory - Definitions and Concepts

Definitions and Concepts

Let V be a vector space over a field F. For instance, suppose V is Rn or Cn, the standard n-dimensional space of column vectors over the real or complex numbers respectively. In this case, the idea of representation theory is to do abstract algebra concretely by using n × n matrices of real or complex numbers.

There are three main sorts of algebraic objects for which this can be done: groups, associative algebras and Lie algebras.

• The set of all invertible n × n matrices is a group under matrix multiplication and the representation theory of groups analyses a group by describing ("representing") its elements in terms of invertible matrices.
• Matrix addition and multiplication make the set of all n × n matrices into an associative algebra and hence there is a corresponding representation theory of associative algebras.
• If we replace matrix multiplication MN by the matrix commutator MN − NM, then the n × n matrices become instead a Lie algebra, leading to a representation theory of Lie algebras.

This generalizes to any field F and any vector space V over F, with linear maps replacing matrices and composition replacing matrix multiplication: there is a group GL(V,F) of automorphisms of V, an associative algebra End_F(V) of all endomorphisms of V, and a corresponding Lie algebra gl(V,F).