Split Casimir operator and universal formulation of the simple Lie algebras

TL;DR

This paper develops characteristic identities for split Casimir operators of simple Lie algebras, enabling explicit invariant projectors in the adjoint representation, and presents a universal formulation using Vogel parameters.

Contribution

It introduces characteristic identities for split Casimir operators and derives explicit invariant projectors in the adjoint representation for all simple Lie algebras, unified through Vogel parameters.

Findings

Explicit formulas for invariant projectors in T^{ ensor 2}

Characteristic identities for split Casimir operators

Universal description of simple Lie algebras via Vogel parameters

Abstract

We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in adjoint representation. By means of these characteristic identities, for all simple Lie algebras we derive explicit formulae for invariant projectors onto irreducible subrepresentations in T^{\otimes 2} in the case when T is the adjoint representation. These projectors and characteristic identities are considered from the viewpoint of the universal description of the simple Lie algebras in terms of the Vogel parameters.