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Difference between revisions of "Subrepresentation of a representation"

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A [[Linear representation|linear representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910201.png" /> in an invariant subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910202.png" /> of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910203.png" /> of a group (algebra, ring or semi-group) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910204.png" /> in a (topological) vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910205.png" /> defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910206.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910207.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910208.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091020/s0910209.png" /> is a [[Continuous representation|continuous representation]] (of a topological group, algebra, ring, or semi-group), then any subrepresentation of it is also continuous.
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A [[Linear representation|linear representation]]  $  \rho $
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in an invariant subspace  $  F \subset  E $
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of a representation  $  \pi $
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of a group (algebra, ring or semi-group) $  X $
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in a (topological) vector space $  E $
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defined by the formula $  \rho ( x) \xi = \pi ( x) \xi $
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for all $  \xi \in F $,  
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$  x \in X $.  
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If $  \pi $
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is a [[Continuous representation|continuous representation]] (of a topological group, algebra, ring, or semi-group), then any subrepresentation of it is also continuous.

Latest revision as of 08:24, 6 June 2020


A linear representation $ \rho $ in an invariant subspace $ F \subset E $ of a representation $ \pi $ of a group (algebra, ring or semi-group) $ X $ in a (topological) vector space $ E $ defined by the formula $ \rho ( x) \xi = \pi ( x) \xi $ for all $ \xi \in F $, $ x \in X $. If $ \pi $ is a continuous representation (of a topological group, algebra, ring, or semi-group), then any subrepresentation of it is also continuous.

How to Cite This Entry:
Subrepresentation of a representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subrepresentation_of_a_representation&oldid=16384
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article