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Difference between revisions of "Linear representation"

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A [[Homomorphism|homomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059420/l0594201.png" /> of a group (respectively an algebra, ring, semi-group) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059420/l0594202.png" /> into the group of all invertible linear operators on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059420/l0594203.png" /> (respectively, into the algebra, ring, multiplicative semi-group of all linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059420/l0594204.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059420/l0594205.png" /> is a topological vector space, then a linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059420/l0594206.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059420/l0594207.png" /> is a representation whose image contains only continuous linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059420/l0594208.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059420/l0594209.png" /> is called the representation space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059420/l05942010.png" /> and the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059420/l05942011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059420/l05942012.png" />, are called the operators of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059420/l05942013.png" />.
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A [[Homomorphism|homomorphism]] $\pi$ of a group (respectively an algebra, ring, semi-group) $X$ into the group of all invertible linear operators on a vector space $E$ (respectively, into the algebra, ring, multiplicative semi-group of all linear operators on $E$). If $E$ is a topological vector space, then a linear representation of $X$ on $E$ is a representation whose image contains only continuous linear operators on $E$. The space $E$ is called the representation space of $\pi$ and the operators $\pi(x)$, $x\in X$, are called the operators of the representation $\pi$.
  
 
====References====
 
====References====

Revision as of 16:07, 30 July 2014

A homomorphism $\pi$ of a group (respectively an algebra, ring, semi-group) $X$ into the group of all invertible linear operators on a vector space $E$ (respectively, into the algebra, ring, multiplicative semi-group of all linear operators on $E$). If $E$ is a topological vector space, then a linear representation of $X$ on $E$ is a representation whose image contains only continuous linear operators on $E$. The space $E$ is called the representation space of $\pi$ and the operators $\pi(x)$, $x\in X$, are called the operators of the representation $\pi$.

References

[1] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)


Comments

References

[a1] C.W. Curtis, I. Reiner, "Methods of representation theory" , 1–2 , Wiley (Interscience) (1981–1987)
[a2] J.-P. Serre, "Répresentations linéaires des groupes finis" , Hermann (1967)
[a3] A.I. Shtern, "Theory of group representations" , Springer (1982) (Translated from Russian)
How to Cite This Entry:
Linear representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_representation&oldid=32577
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article