翻訳と辞書 Words near each other ・ Valvula ・ Valvular heart disease ・ Valvulas Industriales ・ Valvulinidae ・ Valvulitis ・ Valvulotome ・ Valvulotomy ・ Valwig ・ Valwood School ・ Valy ・ Valy (Cheb District) ・ Valy (Pardubice District) ・ Valy Arnheim ・ Valya ・ Valya (singer) ・ Valya algebra ・ Valya Balkanska ・ Valya Dudycz Lupescu ・ Valya Samvelyan ・ Valyakish ・ Valyantsin Byalkevich ・ Valyava ・ Valyavishki Lakes ・ Valyermo, California ・ Valyra (horse) ・ Valyrian languages ・ Valyukevich ・ Valz Prize ・ Valz-sous-Châteauneuf ・ Valzeina 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 Valya algebra ： ウィキペディア英語版 Valya algebra In abstract algebra, a Valya algebra (or Valentina algebra) is a nonassociative algebra ''M'' over a field ''F'' whose multiplicative binary operation ''g'' satisfies the following axioms: 1. The skew-symmetry condition :$g\; (A,\; B)\; =-g\; (B,\; A)$ for all $A,B\; \backslash in\; M$. 2. The Valya identity :$J\; (g\; (A\_1,\; A\_2),\; g\; (A\_3,\; A\_4),\; g\; (A\_5,\; A\_6))\; =0$ for all $A\_k\; \backslash in\; M$, where k=1,2,...,6, and $J\; (A,\; B,\; C):=\; g\; (g\; (A,\; B),\; C)+g\; (g\; (B,\; C),\; A)+g\; (g\; (C,\; A),\; B).$ 3. The bilinear condition :$g(aA+bB,C)=ag(A,C)+bg(B,C)$ for all $A,B,C\; \backslash in\; M$ and $a,b\; \backslash in\; F$. We say that M is a Valya algebra if the commutant of this algebra is a Lie subalgebra. Each Lie algebra is a Valya algebra. There is the following relationship between the commutant-associative algebra and Valentina algebra. The replacement of the multiplication g(A,B) in an algebra M by the operation of commutation ()=g(A,B)-g(B,A), makes it into the algebra $M^$. If M is a commutant-associative algebra, then $M^$ is a Valya algebra. A Valya algebra is a generalization of a Lie algebra. ==Examples== Let us give the following examples regarding Valya algebras. (1) Every finite Valya algebra is the tangent algebra of an analytic local commutant-associative loop (Valya loop) as each finite Lie algebra is the tangent algebra of an analytic local group (Lie group). This is the analog of the classical correspondence between analytic local groups (Lie groups) and Lie algebras. (2) A bilinear operation for the differential 1-forms :$\backslash alpha=F\_k(x)\backslash ,\; dx^k\; ,\; \backslash quad\; \backslash beta=G\_k(x)\backslash ,\; dx^k$ on a symplectic manifold can be introduced by the rule : $(\backslash alpha,\backslash beta)\_0=d\; \backslash Psi(\backslash alpha,\backslash beta)+\; \backslash Psi(d\backslash alpha,\backslash beta)+\backslash Psi(\backslash alpha,d\backslash beta),\; \backslash ,$ where $(\backslash alpha,\backslash beta)$ is 1-form. A set of all nonclosed 1-forms, together with this operation, is Lie algebra. If $\backslash alpha$ and $\backslash beta$ are closed 1-forms, then $d\backslash alpha=d\backslash beta=0$ and : $(\backslash alpha,\backslash beta)=d\; \backslash Psi(\backslash alpha,\backslash beta).\; \backslash ,$ A set of all closed 1-forms, together with this bracket, form a Lie algebra. A set of all nonclosed 1-forms together with the bilinear operation $(\backslash alpha,\backslash beta)$ is a Valya algebra, and it is not a Lie algebra. 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「Valya 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.