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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 Casimir invariant ： ウィキペディア英語版 Casimir element In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the centre of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group. The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931. == Definition == Suppose that $\backslash mathfrak$ is an $n$-dimensional semisimple Lie algebra. Let ''B'' be a bilinear form on $\backslash mathfrak$ that is invariant under the adjoint action of $\backslash mathfrak$ on itself, meaning that $B(ad\_XY,Z)+B(Y,ad\_XZ)=0$ for all X,Y,Z in G. (The most typical choice of ''B'' is the Killing form.) Let :$\backslash \_^n$ be any basis of $\backslash mathfrak$, and :$\backslash \_^n$ be the dual basis of $\backslash mathfrak$ with respect to ''B''. The Casimir element $\backslash Omega$ for ''B'' is the element of the universal enveloping algebra $U(\backslash mathfrak)$ given by the formula :$\backslash Omega\; =\; \backslash sum\_^n\; X\_i\; X^i.$ Although the definition relies on a choice of basis for the Lie algebra, it is easy to show that ''Ω'' is independent of this choice. On the other hand, ''Ω'' does depend on the bilinear form ''B''. The invariance of ''B'' implies that the Casimir element commutes with all elements of the Lie algebra $\backslash mathfrak$, and hence lies in the center of the universal enveloping algebra $U(\backslash mathfrak)$. 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「Casimir element」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.