View source for Completely-reducible set

A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024000/c0240001.png" /> of linear operators on a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024000/c0240002.png" /> with the following property: Any closed subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024000/c0240003.png" /> that is invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024000/c0240004.png" /> has a complement in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024000/c0240005.png" /> that is also invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024000/c0240006.png" />. In a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024000/c0240007.png" /> any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024000/c0240008.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024000/c0240009.png" /> of an algebra (group, ring, etc.) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024000/c02400010.png" /> is called completely reducible if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024000/c02400011.png" /> is completely reducible. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024000/c02400012.png" /> is a compact group or a semi-simple connected Lie group (Lie algebra), any representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024000/c02400013.png" /> in a finite-dimensional vector space is completely reducible (the principle of complete reducibility).

====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>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.E. Humphreys,   "Introduction to Lie algebras and representation theory" , Springer  (1972)</TD></TR></table>