topological group representation

1 Finite Dimensional Representations

Let $G$ be a topological group and $V$ a finite-dimensional normed vector space. We denote by $GL(V)$ the general linear group of $V$, endowed with the topology coming from the operator norm.

Regarding only the group structure of $G$, recall that a representation of $G$ in $V$ is a group homomorphism $\pi :G⟶GL(V)$.

$$

Definition - A representation of the topological group $G$ in $V$ is a continuous group homomorphism $\pi :G⟶GL(V)$, i.e. is a continuous representation of the abstract group $G$ in $V$.

$$

We have the following equivalent definitions:

• A representation of $G$ in $V$ is a group homomorphism $\pi :G⟶GL(V)$ such that the mapping $G\times V⟶V$ defined by$(g,v)\mapsto \pi (g)v$is continuous.

• A representation of $G$ in $V$ is a group homomorphism $\pi :G⟶GL(V)$ such that, for every $v\in V$, the mapping $G⟶V$ defined by$g\mapsto \pi (g)v$is continuous.

2 Representations in Hilbert Spaces

Let $G$ be a topological group and $H$ a Hilbert space. We denote by $B(H)$ the algebra of bounded operators endowed with the strong operator topology (this topology does not coincide with the norm topology unless $H$ is finite-dimensional). Let $𝒢(H)$ the set of invertible operators in $B(H)$ endowed with the subspace topology.

$$

Definition - A representation of the topological group $G$ in $H$ is a continuous group homomorphism $\pi :G⟶𝒢(H)$, i.e. is a continuous representation of the abstract group $G$ in $H$.

$$

We denote by $rep(G,H)$ the set of all representations of $G$ in the Hilbert space $H$.

We have the following equivalent definitions:

• A representation of $G$ in $H$ is a group homomorphism $\pi :G⟶𝒢(H)$ such that the mapping $G\times H⟶H$ defined by$(g,v)\mapsto \pi (g)v$is continuous.

• A representation of $G$ in $H$ is a group homomorphism $\pi :G⟶𝒢(H)$ such that, for every $v\in H$, the mapping $G⟶H$ defined by$g\mapsto \pi (g)v$is continuous.

Remark - The 3rd definition is exactly the same as the 1st definition, just written in other .

3 Representations as G-modules

Recall that, for an abstract group $G$, it is the same to consider a representation of $G$ or to consider a $G$-module (http://planetmath.org/GModule), i.e. to each representation of $G$ corresponds a $G$-module and vice-versa.

For a topological group $G$, representations of $G$ satisfy some continuity . Thus, we are not interested in all $G$-modules, but rather in those which are compatible with the continuity conditions.

$$

Definition - Let $G$ be a topological group. A $G$-module is a normed vector space (or a Hilbert space) $V$ where $G$ acts continuously, i.e. there is a continuous action $\psi :G\times V⟶V$.

$$

To give a representation of a topological group $G$ is the same as giving a $G$-module (in the sense described above).

4 Special Kinds of Representations

• Let $\pi \in rep(G,H)$. We say that a subspace $V\subseteq H$ is by $\pi$ if $V$ is invariant under every operator $\pi (s)$ with $s\in G$.

• A of a representation $\pi \in rep(G,H)$ is a representation ${\pi }_0\in rep(G,H_0)$ obtained from $\pi$ by restricting to a closed subspace $H_0\subseteq H$.

• A representation $\pi \in rep(G,H)$ is said to be if the only closed subspaces of $H$ are the trivial ones, $\{0\}$ and $H$.

• Two representations ${\pi }_1:G⟶GL(V_1)$ and ${\pi }_2:G⟶GL(V_2)$ of a topological group $G$ are said to be equivalent if there exists an invertible linear transformation $T:V_1⟶V_2$ such that for every $g\in G$ one has ${\pi }_1(g)=T^{-1}{\pi }_2(g)T$.

  The definition is similar for Hilbert spaces, by taking $T$ as an invertible bounded linear operator.