# Completely-reducible set

A set of linear operators on a topological vector space with the following property: Any closed subspace in that is invariant with respect to has a complement in that is also invariant with respect to . In a Hilbert space any set 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 of an algebra (group, ring, etc.) is called completely reducible if the set is completely reducible. If is a compact group or a semi-simple connected Lie group (Lie algebra), any representation of 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. D.P. Zhelobenko (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Completely-reducible_set&oldid=19183