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.

Group action perspective

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

$Φ : G \times V ⟶ V$

such that for a fixed $g$ , the map

φ_{g} : V v ⟶ V ⟼ Φ (g, v) = g \cdot v

is linear.

Group homomorphims perspective

Another perspective is that a representation is a smooth group homomorphism

$Φ : G ⟶ GL (V) .$

Morphisms

For representations $V$ and $W$ , a morphism is a linear map $f : V \to W$ that is equivariant, i.e.

$f (g \cdot v) = g \cdot f (v) \forall g \in G, v \in V .$

We denote the set of all morphisms $Hom_{G} (V, W)$ .

Note $Hom_{G} (V, W) \subset Hom (V, W)$ is a linear subspace.

Subrepresentations

For a representation $G \to V$ , a linear subspace $U \subset V$ is a subrepresentation if it is $G$ -invariant. That is if

$g \cdot u \in U \forall g \in G, u \in U .$

Why is a representation a “module”?

todo

Building new representations from old ones

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

Direct sum

$V \oplus W$ is a representation with action $g (v, w) = (gv, g w)$

If both are matrix representations, with matrices $g \mapsto A (g)$ and $g \mapsto B (g)$ then the new matrix representation will be the matrix

g ⟼ [A (g) 0 0 B (g)]

Tensor product

The tensor product of two representations gives a representation:

g (v \otimes w) = gv \otimes g w .

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

Morphims between representations

$G$ acts on $Hom (V, W)$ by

(g \cdot f) (v) = g ∙_{W} f (g^{- 1} ∙_{V} v) .

This gives the following diagram

Note that if $W = C$ (or $R$ depending on base field) and the action of $G$ on $C$ is trivial we get the dual representation

(g \cdot φ) (v) = g \cdot φ (g^{- 1} (v)) = φ (g^{- 1} (V)) \forall φ \in V^{*}, g \in G

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Lie group representation