Difference between revisions of "Quotient representation"
From Encyclopedia of Mathematics
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| − | A quotient representation of a representation | + | {{TEX|done}} |
| + | 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
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