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In mathematics, a free Lie algebra, over a given field ''K'', is a Lie algebra generated by a set ''X'', without any imposed relations. ==Definition== : Let ''X'' be a set and ''i'': ''X'' → ''L'' a morphism of sets from ''X'' into a Lie algebra ''L''. The Lie algebra ''L'' is called free on ''X'' if for any Lie algebra ''A'' with a morphism of sets ''f'': ''X'' → ''A'', there is a unique Lie algebra morphism ''g'': ''L'' → ''A'' such that ''f'' = ''g'' o ''i''. Given a set ''X'', one can show that there exists a unique free Lie algebra ''L(X)'' generated by ''X''. In the language of category theory, the functor sending a set ''X'' to the Lie algebra generated by ''X'' is the free functor from the category of sets to the category of Lie algebras. That is, it is left adjoint to the forgetful functor. The free Lie algebra on a set ''X'' is naturally graded. The 0-graded component of the free Lie algebra is just the free vector space on that set. One can alternatively define a free Lie algebra on a vector space ''V'' as left adjoint to the forgetful functor from Lie algebras over a field ''K'' to vector spaces over the field ''K'' – forgetting the Lie algebra structure, but remembering the vector space structure. 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「Free Lie algebra」の詳細全文を読む スポンサード リンク
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