# Difference between revisions of "Linear representation"

From Encyclopedia of Mathematics

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− | A [[Homomorphism|homomorphism]] | + | {{TEX|done}} |

+ | 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=16731

This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article