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

TL;DR

This paper develops characteristic identities and projectors for the 3-split Casimir operators of simple Lie algebras in the adjoint representation, providing universal formulas involving Vogel parameters for subrepresentations in the tensor cube.

Contribution

It introduces a universal framework for analyzing the 3-split Casimir operators and subrepresentations in the tensor cube of simple Lie algebras using Vogel parameters.

Findings

Derived characteristic identities for 3-split Casimir operators.

Constructed projectors onto invariant subspaces from these identities.

Provided universal formulas for traces and dimensions of subrepresentations.

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.