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In mathematics, Schur's lemma〔Issai Schur (1905) ("Neue Begründung der Theorie der Gruppencharaktere" ) (New foundation for the theory of group characters), ''Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin'', pages 406-432.〕 is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a group ''G'' and ''φ'' is a linear map from ''M'' to ''N'' that commutes with the action of the group, then either ''φ'' is invertible, or ''φ'' = 0. An important special case occurs when ''M'' = ''N'' and ''φ'' is a self-map. The lemma is named after Issai Schur who used it to prove Schur orthogonality relations and develop the basics of the representation theory of finite groups. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which is due to Jacques Dixmier. == Formulation in the language of modules == If ''M'' and ''N'' are two simple modules over a ring ''R'', then any homomorphism ''f'': ''M'' → ''N'' of ''R''-modules is either invertible or zero. In particular, the endomorphism ring of a simple module is a division ring.〔Lam (2001), (p. 33 ).〕 The condition that ''f'' is a module homomorphism means that : The group version is a special case of the module version, since any representation of a group ''G'' can equivalently be viewed as a module over the group ring of ''G''. Schur's lemma is frequently applied in the following particular case. Suppose that ''R'' is an algebra over a field ''k'' and the vector space ''M'' = ''N'' is a simple module of ''R''. Then Schur's lemma says that the endomorphism ring of the module ''M'' is a division algebra over the field ''k''. If ''M'' is finite-dimensional, this division algebra is finite-dimensional. If ''k'' is the field of complex numbers, the only option is that this division algebra is the complex numbers. Thus the endomorphism ring of the module ''M'' is "as small as possible". In other words, the only linear transformations of ''M'' that commute with all transformations coming from ''R'' are scalar multiples of the identity. This holds more generally for any algebra ''R'' over an uncountable algebraically closed field ''k'' and for any simple module ''M'' that is at most countably-dimensional: the only linear transformations of ''M'' that commute with all transformations coming from ''R'' are scalar multiples of the identity. When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is still of particular interest. A simple module over ''k''-algebra is said to be absolutely simple if its endomorphism ring is isomorphic to ''k''. This is in general stronger than being irreducible over the field ''k'', and implies the module is irreducible even over the algebraic closure of ''k''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Schur's lemma」の詳細全文を読む スポンサード リンク
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