翻訳と辞書 Words near each other ・ Lie (disambiguation) ・ Lie (song) ・ Lie (surname) ・ Lie (T-ara song) ・ Lie a Little Better ・ Lie About Us ・ Lie algebra ・ Lie algebra bundle ・ Lie algebra cohomology ・ Lie algebra extension ・ Lie algebra representation ・ Lie algebra-valued differential form ・ Lie algebroid ・ Lie Back and Enjoy It ・ Lie Back and Think of England ・ Lie bialgebra ・ Lie bialgebroid ・ Lie bracket of vector fields ・ Lie Cliff ・ Lie coalgebra ・ Lie conformal algebra ・ Lie derivative ・ Lie detection ・ Lie Detector (disambiguation) ・ Lie Detector (TV series) ・ Lie Detectors ・ Lie Die ・ Lie Down in Darkness ・ Lie Down in Darkness (Moby song) ・ Lie Down in Darkness (novel) Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Lie bialgebra ： ウィキペディア英語版 Lie bialgebra In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it's a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the comultiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1-cocycle, so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary. They are also called Poisson-Hopf algebras, and are the Lie algebra of a Poisson-Lie group. Lie bialgebras occur naturally in the study of the Yang-Baxter equations. ==Definition== A vector space $\backslash mathfrak$ is a Lie bialgebra if it is a Lie algebra, and there is the structure of Lie algebra also on the dual vector space $\backslash mathfrak^$ * which is compatible. More precisely the Lie algebra structure on $\backslash mathfrak$ is given by a Lie bracket $(,\backslash ):\backslash mathfrak\; \backslash otimes\; \backslash mathfrak\; \backslash to\; \backslash mathfrak$ and the Lie algebra structure on $\backslash mathfrak^$ * is given by a Lie bracket $\backslash delta^$ *:\mathfrak^ * \otimes \mathfrak^ * \to \mathfrak^ *. Then the map dual to $\backslash delta^$ * is called the cocommutator, $\backslash delta:\backslash mathfrak\; \backslash to\; \backslash mathfrak\; \backslash otimes\; \backslash mathfrak$ and the compatibility condition is the following cocyle relation: :$\backslash delta(())\; =\; \backslash left($ \operatorname_X \otimes 1 + 1 \otimes \operatorname_X \right) \delta(Y) - \left( \operatorname_Y \otimes 1 + 1 \otimes \operatorname_Y \right) \delta(X) where $\backslash operatorname\_XY=()$ is the adjoint. Note that this definition is symmetric and $\backslash mathfrak^$ * is also a Lie bialgebra, the dual Lie bialgebra. 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 bialgebra」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.