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Casimir element : 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 \mathfrak is an n-dimensional semisimple Lie algebra. Let ''B'' be a bilinear form on \mathfrak that is invariant under the adjoint action of \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
:\_^n
be any basis of \mathfrak, and
:\_^n
be the dual basis of \mathfrak with respect to ''B''. The Casimir element \Omega for ''B'' is the element of the universal enveloping algebra U(\mathfrak) given by the formula
:\Omega = \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 \mathfrak, and hence lies in the center of the universal enveloping algebra U(\mathfrak).

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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