# 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 $\pi:G\longrightarrow GL(V)$.

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

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We have the following equivalent definitions:

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

• A representation of $G$ in $V$ is a group homomorphism $\pi:G\longrightarrow GL(V)$ such that, for every $v\in V$, the mapping $G\longrightarrow V$ 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 $\mathcal{G}(H)$ the set of invertible operators in $B(H)$ endowed with the subspace topology.

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

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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 $\pi:G\longrightarrow\mathcal{G}(H)$ such that the mapping $G\times H\longrightarrow H$ defined by $(g,v)\mapsto\pi(g)v\;$ is continuous.

• A representation of $G$ in $H$ is a group homomorphism $\pi:G\longrightarrow\mathcal{G}(H)$ such that, for every $v\in H$, the mapping $G\longrightarrow H$ 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.

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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 $\psi:G\times V\longrightarrow V$.

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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 $\pi_{1}:G\longrightarrow GL(V_{1})$ and $\pi_{2}:G\longrightarrow GL(V_{2})$ of a topological group $G$ are said to be equivalent if there exists an invertible linear transformation $T:V_{1}\longrightarrow V_{2}$ 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.

Title topological group representation TopologicalGroupRepresentation 2013-03-22 18:02:18 2013-03-22 18:02:18 asteroid (17536) asteroid (17536) 8 asteroid (17536) Definition msc 43A65 msc 22A25 msc 22A05 representation of topological groups equivalent representations of topological groups