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).


[1] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)


The principle of complete reducibility is commonly referred to as Weyl's theorem (cf. [a1], Chapt. 2 Sect. 6).


[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:
This article was adapted from an original article by D.P. Zhelobenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article