Split Casimir operator for simple Lie algebras in the cube of ad-representation and Vogel parameters

Abstract

In this paper, we constructed characteristic identities for the 3-split (polarized) Casimir operators of simple Lie algebras in the adjoint representations [Formula: see text] and deduced a certain class of subrepresentations in [Formula: see text]. 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 [Formula: see text] were calculated. All our formulas are in agreement with the universal description of (irreducible) subrepresentations in [Formula: see text] for simple Lie algebras in terms of the Vogel parameters.