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Difference between revisions of "Quotient representation"

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A quotient representation of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q0769101.png" /> of a group (cf. [[Representation of a group|Representation of a group]]), or algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q0769102.png" /> is a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q0769103.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q0769104.png" /> defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q0769105.png" /> be the (topological) vector space of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q0769106.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q0769107.png" /> is a representation in a (topological) vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q0769108.png" /> that is the quotient space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q0769109.png" /> by some invariant subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q07691010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q07691011.png" /> (cf. [[Invariant subspace of a representation|Invariant subspace of a representation]]), defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q07691012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q07691013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q07691014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076910/q07691015.png" /> is a [[Continuous representation|continuous representation]], then so is any quotient representation of it.
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A quotient representation of a representation $\pi$ of a group (cf. [[Representation of a group|Representation of a group]]), or algebra, $X$ is a representation $\rho$ of $X$ defined as follows. Let $E$ be the (topological) vector space of the representation $\pi$; then $\rho$ is a representation in a (topological) vector space $E/F$ that is the quotient space of $E$ by some invariant subspace $F$ of $\pi$ (cf. [[Invariant subspace of a representation|Invariant subspace of a representation]]), defined by the formula $\rho(x)(\xi+F)=\pi(x)\xi+F$ for all $x\in X$, $\xi\in E$. If $\pi$ is a [[Continuous representation|continuous representation]], then so is any quotient representation of it.

Latest revision as of 15:40, 22 July 2014

A quotient representation of a representation $\pi$ of a group (cf. Representation of a group), or algebra, $X$ is a representation $\rho$ of $X$ defined as follows. Let $E$ be the (topological) vector space of the representation $\pi$; then $\rho$ is a representation in a (topological) vector space $E/F$ that is the quotient space of $E$ by some invariant subspace $F$ of $\pi$ (cf. Invariant subspace of a representation), defined by the formula $\rho(x)(\xi+F)=\pi(x)\xi+F$ for all $x\in X$, $\xi\in E$. If $\pi$ is a continuous representation, then so is any quotient representation of it.

How to Cite This Entry:
Quotient representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quotient_representation&oldid=32502
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article