拟单正交代数完备族的Casimir不变量
Casimir invariants for the complete family of quasi-simple orthogonal algebras
摘要：
在统一的条件下，得到了任意维数的正交型实拟单李代数的包络代数的中心生成元的完全选择。这些结果同时包含了伪正交代数$so(p,q)$的多项式不变量，以及非齐次$iso(p,q)$、Newton-Hooke和Galilei型等非单代数的Casimirs不变量，其通过从单代数$so(p,q)$开始的收缩而获得。一个拟单正交代数的包络代数的中心维数与它们通过收缩而得的单$so(p,q)$代数的包络代数的中心维数相同。高阶不变量的结构给出了一个方便的“金字塔”的方式，在某些套“保利-鲁班斯基”元素的包络代数。作为一个例子，该方案被应用于恢复（3+1）运动学代数的Casimirs。对这些结果与展开式研究的相关性也作了一些展望。
英文摘要：
A complete choice of generators of the center of the enveloping algebras of real quasi-simple Lie algebras of orthogonal type, for arbitrary dimension, is obtained in a unified setting. The results simultaneously include the well known polynomial invariants of the pseudo-orthogonal algebras $so(p,q)$, as well as the Casimirs for many non-simple algebras such as the inhomogeneous $iso(p,q)$, the Newton-Hooke and Galilei type, etc., which are obtained by contraction(s) starting from the simple algebras $so(p,q)$. The dimension of the center of the enveloping algebra of a quasi-simple orthogonal algebra turns out to be the same as for the simple $so(p,q)$ algebras from which they come by contraction. The structure of the higher order invariants is given in a convenient "pyramidal" manner, in terms of certain sets of "Pauli-Lubanski" elements in the enveloping algebras. As an example showing this approach at work, the scheme is applied to recovering the Casimirs for the (3+1) kinematical algebras. Some prospects on the relevance of these results for the study of expansions are also given.
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