If are two algebraically-irreducible representations of some group or algebra in two vector spaces and , respectively, then any intertwining operator for the representations and is either zero or provides a one-to-one mapping from onto (in this case and are equivalent). The lemma was established by I. Schur
for finite-dimensional irreducible representations. The description of the family of intertwining operators for two given representations is an analogue of the Schur lemma. In particular, the following statement is often called Schur's lemma: If and are unitary irreducible representations of some group or are symmetric irreducible representations of some algebra in two Hilbert spaces and , respectively, then any closed linear operator from into intertwining and is either zero or unitary (in this case and are unitarily equivalent). The description of the family of intertwining operators for representations that allow for an expansion in a direct integral is called the continuous analogue of Schur's lemma.
The two following statements are generalizations of Schur's lemma to families of operators acting on infinite-dimensional spaces.
Let be two representations in Hilbert spaces and of a symmetric ring . Let be a closed linear operator with zero kernel and dense domain and range. If the relations hold for all , then the representations and are unitarily equivalent.
Let be an algebra of continuous linear operators in a locally convex space containing a non-zero compact operator and having no non-trivial closed invariant subspaces. Then any operator permutable with all operators from is a multiple of the identity operator.
|||I. Schur, "Arithmetische Untersuchungen über endliche Gruppen linearer Substitutionen" Sitzungsber. Akad. Wiss. Berlin (1906) pp. 164–184|
|||A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)|
|||M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)|
|||A.I. Shtern, "Theory of group representations" , Springer (1982) (Translated from Russian)|
|||D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)|
|||V.I. Lomonosov, "Invariant subspaces for the family of operators which commute with a completely continuous operator" Funct. Anal. Appl. , 7 : 3 (1973) pp. 213–214 Funktsional. Anal. i Prilozhen. , 7 : 3 (1973) pp. 55–56|
The Schur lemma has a number of immediate consequences. An important one is that if is an algebraically-irreducible representation in a linear space over a field , then the set of intertwining operators of is a skew-field over . If , this means that , i.e. every intertwining operator is a multiple of the identity. If , this means that , or , the -algebra of quaternions.
|[a1]||C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) pp. §44|
|[a2]||J.-P. Serre, "Linear representations of finite groups" , Springer (1982) (Translated from French)|
|[a3]||C.E. Rickart, "General theory of Banach algebras" , v. Nostrand (1960)|
|[a4]||B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) pp. 64|
|[a5]||N. Bourbaki, "Algèbre" , Eléments de mathématiques , Hermann (1958) pp. Chapt. 8. Modules et anneaux semi-simples|
Schur lemma. A.I. Shtern, V.I. Lomonosov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Schur_lemma&oldid=17919