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In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra. In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays a decisive role. The universality of this construction of this ring says that the category of representations of a Lie algebra is the same as the category of modules over its enveloping algebra. ==Formal definition== A representation of a Lie algebra is a Lie algebra homomorphism : from to the Lie algebra of endomorphisms on a vector space ''V'' (with the commutator as the Lie bracket), sending an element ''x'' of to an element ''ρ''''x'' of . Explicitly, this means that : for all ''x,y'' in . The vector space ''V'', together with the representation ρ, is called a -module. (Many authors abuse terminology and refer to ''V'' itself as the representation). The representation is said to be faithful if it is injective. One can equivalently define a -module as a vector space ''V'' together with a bilinear map such that : for all ''x,y'' in and ''v'' in ''V''. This is related to the previous definition by setting ''x'' ⋅ ''v'' = ρ''x'' (v). 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「Lie algebra representation」の詳細全文を読む スポンサード リンク
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