Difference between revisions of "Completely-reducible set"
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| + | A set $ M $ | ||
| + | of linear operators on a topological vector space $ E $ | ||
| + | with the following property: Any closed subspace in $ E $ | ||
| + | that is invariant with respect to $ M $ | ||
| + | has a complement in $ E $ | ||
| + | that is also invariant with respect to $ M $. | ||
| + | In a Hilbert space $ E $ | ||
| + | any set $ M $ | ||
| + | that is symmetric with respect to Hermitian conjugation is completely reducible. In particular, any group of unitary operators is a completely-reducible set. A representation $ \phi $ | ||
| + | of an algebra (group, ring, etc.) $ A $ | ||
| + | is called completely reducible if the set $ M = \{ {\phi (a) } : {a \in A } \} $ | ||
| + | is completely reducible. If $ A $ | ||
| + | is a compact group or a semi-simple connected Lie group (Lie algebra), any representation of $ A $ | ||
| + | in a finite-dimensional vector space is completely reducible (the principle of complete reducibility). | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
Latest revision as of 16:40, 31 March 2020
A set $ M $
of linear operators on a topological vector space $ E $
with the following property: Any closed subspace in $ E $
that is invariant with respect to $ M $
has a complement in $ E $
that is also invariant with respect to $ M $.
In a Hilbert space $ E $
any set $ M $
that is symmetric with respect to Hermitian conjugation is completely reducible. In particular, any group of unitary operators is a completely-reducible set. A representation $ \phi $
of an algebra (group, ring, etc.) $ A $
is called completely reducible if the set $ M = \{ {\phi (a) } : {a \in A } \} $
is completely reducible. If $ A $
is a compact group or a semi-simple connected Lie group (Lie algebra), any representation of $ A $
in a finite-dimensional vector space is completely reducible (the principle of complete reducibility).
References
| [1] | D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) |
Comments
The principle of complete reducibility is commonly referred to as Weyl's theorem (cf. [a1], Chapt. 2 Sect. 6).
References
| [a1] | J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) |
How to Cite This Entry:
Completely-reducible set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Completely-reducible_set&oldid=19183
Completely-reducible set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Completely-reducible_set&oldid=19183
This article was adapted from an original article by D.P. Zhelobenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article