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In mathematics, a Manin triple (''g'', ''p'', ''q'') consists of a Lie algebra ''g'' with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras ''p'' and ''q'' such that ''g'' is the direct sum of ''p'' and ''q'' as a vector space. Manin triples were introduced by , who named them after Yuri Manin. classified the Manin triples where ''g'' is a complex reductive Lie algebra. ==Manin triples and Lie bialgebras== If (''g'', ''p'', ''q'') is a finite-dimensional Manin triple then ''p'' can be made into a Lie bialgebra by letting the cocommutator map ''p'' → ''p'' ⊗ ''p'' be dual to the map ''q'' ⊗ ''q'' → ''q'' (using the fact that the symmetric bilinear form on ''g'' identifies ''q'' with the dual of ''p''). Conversely if ''p'' is a Lie bialgebra then one can construct a Manin triple from it by letting ''q'' be the dual of ''p'' and defining the commutator of ''p'' and ''q'' to make the bilinear form on ''g'' = ''p'' ⊕ ''q'' invariant. 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「Manin triple」の詳細全文を読む スポンサード リンク
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