Difference between revisions of "Analytic representation"
From Encyclopedia of Mathematics
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| − | A representation of a complex Lie group | + | A representation of a complex Lie group $G$ in a topological vector space $E$ in which all matrix elements $(\phi(g)\xi,\eta)$, $\xi\in E$, $\eta\in E'$, where $E'$ is the dual topological vector space, are holomorphic on $G$. A representation $\phi$ is called an anti-analytic representation if its matrix elements become holomorphic after complex conjugation. An analytic (anti-analytic) representation of a connected Lie group is uniquely determined by a corresponding Lie algebra representation of this group (cf. [[Representation of a Lie algebra|Representation of a Lie algebra]]). If $G$ is a semi-simple complex Lie group, then all its topologically irreducible analytic (anti-analytic) representations are finite-dimensional. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representation" , Amer. Math. Soc. (1973) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representation" , Amer. Math. Soc. (1973) (Translated from Russian)</TD></TR></table> | ||
Latest revision as of 14:17, 30 July 2014
holomorphic representation
A representation of a complex Lie group $G$ in a topological vector space $E$ in which all matrix elements $(\phi(g)\xi,\eta)$, $\xi\in E$, $\eta\in E'$, where $E'$ is the dual topological vector space, are holomorphic on $G$. A representation $\phi$ is called an anti-analytic representation if its matrix elements become holomorphic after complex conjugation. An analytic (anti-analytic) representation of a connected Lie group is uniquely determined by a corresponding Lie algebra representation of this group (cf. Representation of a Lie algebra). If $G$ is a semi-simple complex Lie group, then all its topologically irreducible analytic (anti-analytic) representations are finite-dimensional.
References
| [1] | M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) |
| [2] | D.P. Zhelobenko, "Compact Lie groups and their representation" , Amer. Math. Soc. (1973) (Translated from Russian) |
How to Cite This Entry:
Analytic representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_representation&oldid=32564
Analytic representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_representation&oldid=32564
This article was adapted from an original article by D.P. Zhelobenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article