Split Casimir operator for simple Lie algebras in the cube of $\mathsf{ad}$-representation and Vogel parameters

Abstract

We constructed characteristic identities for the 3-split (polarized) Casimir operators of simple Lie algebras in the adjoint representations $\mathsf{ad}$ and deduced a certain class of subrepresentations in ${\mathsf{ad}}^{\otimes 3}$. The projectors onto invariant subspaces for these subrepresentations were directly constructed from the characteristic identities for the 3-split Casimir operators. For all simple Lie algebras, universal expressions for the traces of higher powers of the 3-split Casimir operators were found and dimensions of the subrepresentations in ${\mathsf{ad}}^{\otimes 3}$ were calculated. All our formulas are in agreement with the universal description of (irreducible) subrepresentations in ${\mathsf{ad}}^{\otimes 3}$ for simple Lie algebras in terms of the Vogel parameters.

References in corpus (1)

• The split Casimir operator and solutions of the Yang-Baxter equation for the $osp(M|N)$ and $s\ell (M|N)$ Lie superalgebras, higher Casimir operators, and the Vogel parameters

Cited by in corpus (7)

• Macdonald deformation of Vogel's universality and link hyperpolynomials
• On Refined Vogel's universality
• Vogel's universality and Macdonald dimensions
• Can Yang-Baxter imply Lie algebra?
• Torus knots in adjoint representation and Vogel's universality
• Vogel's universality and the classification problem for Jacobi identities
• Universal quantum dimensions: $\gamma$-dependent factors