# Completely-reducible set

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=44954