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Difference between revisions of "Analytic representation"

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''holomorphic representation''
 
''holomorphic representation''
  
A representation of a complex Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123901.png" /> in a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123902.png" /> in which all matrix elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123904.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123905.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123906.png" /> is the dual topological vector space, are holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123907.png" />. A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123908.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123909.png" /> is a semi-simple complex Lie group, then all its topologically irreducible analytic (anti-analytic) representations are finite-dimensional.
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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
This article was adapted from an original article by D.P. Zhelobenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article