On Universal Eigenvalues of Casimir Operator

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 adjoint representations of simple Lie algebras can be represented in a universal form. We show that it complies with $N⟶-N$ duality of the same operator for $SO(2n)$ and $Sp(2n)$ algebras (the part of $N\leftrightarrow -N$ duality of gauge $SO(2n)$ and $Sp(2n)$ theories). We discuss the phenomena of non-zero universal values of Casimir operator on zero representations.

13 pages, 6 tables. arXiv admin note: text overlap with arXiv:1812.07914

References in corpus (1)

• $X_2$ series of universal quantum dimensions

Cited by in corpus (1)

• Vogel's universality and the classification problem for Jacobi identities