Abstract

Motivated by the universal knot polynomials in the gauge Chern–Simons theory, we show that the values of the second Casimir operator on an arbitrary power of Cartan product of \({{X}_{2}}\) and the adjoint representations of simple Lie algebras can be represented in a universal form. We show that it complies with the Perelomov–Popov formula for the generating function for the Casimir spectra. We discuss the phenomenon of non-zero universal values of the Casimir operator on zero representations.

中文翻译：

---

关于卡西米尔算子的通用特征值

摘要

根据规范Chern–Simons理论中的通用结多项式，我们证明了第二个Casimir算子的值是\（{{X} _ {2}} \）的Cartan乘积的任意幂和的伴随表示。简单的李代数可以以通用形式表示。我们证明它符合卡西米尔光谱的生成函数的Perelomov–Popov公式。我们讨论了零表示形式下卡西米尔算子的非零通用值的现象。