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.

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• Vogel's universality and the classification problem for Jacobi identities
• Universal quantum dimensions: $\gamma$-dependent factors