Overview

A Lie group representation is a special case of a group representation.

Definition

There are two ways to define/ think about Lie group representations.

A representation of a Lie group on a vector space is a continuous action

such that for a fixed , the map

is linear.

Another perspective is that a representation is a smooth group homomorphism

For representations and , a morphism is a linear map that is equivariant, i.e.

We denote the set of all morphisms .

Note is a linear subspace.

For a representation , a linear subspace is a subrepresentation if it is -invariant. That is if

Given a Lie group with representations and , we can use algebraic constructions to build new representations.

is a representation with action

If both are matrix representations, with matrices and then the new matrix representation will be the matrix

The tensor product of two representations gives a representation:

In explicit matrix forms: … do this in a sec.

acts on by

This gives the following diagram

Note that if (or depending on base field) and the action of on is trivial we get the dual representation