# 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)