topological group representation
1 Finite Dimensional Representations
Let $G$ be a topological group^{} and $V$ a finitedimensional 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^{} $\pi :G\u27f6GL(V)$.
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Definition  A representation of the topological group $G$ in $V$ is a continuous group homomorphism^{} $\pi :G\u27f6GL(V)$, i.e. is a continuous representation of the abstract group $G$ in $V$.
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We have the following equivalent^{} definitions:

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A representation of $G$ in $V$ is a group homomorphism $\pi :G\u27f6GL(V)$ such that the mapping $G\times V\u27f6V$ defined by $(g,v)\mapsto \pi (g)v$ is continuous^{}.

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A representation of $G$ in $V$ is a group homomorphism $\pi :G\u27f6GL(V)$ such that, for every $v\in V$, the mapping $G\u27f6V$ 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 finitedimensional). 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 $\pi :G\u27f6\mathcal{G}(H)$, 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:

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A representation of $G$ in $H$ is a group homomorphism $\pi :G\u27f6\mathcal{G}(H)$ such that the mapping $G\times H\u27f6H$ defined by $(g,v)\mapsto \pi (g)v$ is continuous.

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A representation of $G$ in $H$ is a group homomorphism $\pi :G\u27f6\mathcal{G}(H)$ such that, for every $v\in H$, the mapping $G\u27f6H$ 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 Gmodules
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 viceversa.
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 $\psi :G\times V\u27f6V$.
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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
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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 ${H}_{0}\subseteq H$.
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A representation $\pi \in rep(G,H)$ is said to be if the only closed subspaces of $H$ are the trivial ones, $\{0\}$ and $H$.

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Two representations ${\pi}_{1}:G\u27f6GL({V}_{1})$ and ${\pi}_{2}:G\u27f6GL({V}_{2})$ of a topological group $G$ are said to be equivalent if there exists an invertible linear transformation $T:{V}_{1}\u27f6{V}_{2}$ 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 

Canonical name  TopologicalGroupRepresentation 
Date of creation  20130322 18:02:18 
Last modified on  20130322 18:02:18 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  8 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 43A65 
Classification  msc 22A25 
Classification  msc 22A05 
Synonym  representation of topological groups 
Defines  equivalent representations of topological groups 