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.

arXiv comment13 pages, 6 tables. arXiv admin note: text overlap with arXiv:1812.07914

References in corpus (10)

• Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots
• On universal knot polynomials
• Universality in Chern-Simons theory
• Nonperturbative universal Chern-Simons theory
• Casimir eigenvalues for universal Lie algebra
• On duality and negative dimensions in the theory of Lie groups and symmetric spaces
• On Universal Quantum Dimensions
• Universal volume of groups and anomaly of Vogel's symmetry
• Partition function of Chern-Simons theory as renormalized q-dimension
• $X_2$ series of universal quantum dimensions

Cited by in corpus (1)

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