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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Differential graded Lie algebra ： ウィキペディア英語版 Differential graded Lie algebra In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible in some sense. They have applications in deformation theory〔Hinich, DG coalgebras as formal stacks 〕 and rational homotopy theory. == Definition == A differential graded Lie algebra is a graded vector space $L\; =\; \backslash bigoplus\; L\_i$ over a field of characteristic zero together with a bilinear map $():\; L\_i\; \backslash otimes\; L\_j\; \backslash to\; L\_$ and a differential $d:\; L\_i\; \backslash to\; L\_$ satisfying :$()\; =\; (-1)^(),$ the graded Jacobi identity: :$(-1)^]\; +(-1)^\; =\; (x,y)\; +\; (-1)^(d\; y)$ for any homogeneous elements ''x'', ''y'' and ''z'' in ''L''. Notice here that the differential lowers the degree and so this differential graded Lie algebra is considered to be homologically graded. If instead the differential raised degree the differential graded Lie algebra is said to be cohomologically graded (usually to reinforce this point the grading is written in superscript: $L^i$). The choice of (co)homological grading usually depends upon personal preference or the situation as they are equivalent: a homologically graded space can be made into a cohomological one via setting $L^i=L\_$. Alternative equivalent definitions of a differential graded Lie algebra include: # a Lie algebra object internal to the category of chain complexes; # a strict $L\_\backslash infty$-algebra. A morphism of differential graded Lie algebras is a graded linear map $f:L\backslash to\; L^\backslash prime$ that commutes with the bracket and the differential, i.e. $f\; ()\_=()\_$ and $f\; (d\_L\; x)\; =\; d\_\; f\; (x)$. Differential graded Lie algebras and their morphisms define a category. 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「Differential graded Lie algebra」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.