Split Casimir operator for simple Lie algebras, solutions of Yang-Baxter equations and Vogel parameters

TL;DR

This paper develops characteristic identities for split Casimir operators of simple Lie algebras, enabling explicit construction of invariant projectors and solutions to the Yang-Baxter equations, with a universal perspective via Vogel parameters.

Contribution

It introduces characteristic identities for split Casimir operators and derives explicit invariant projectors for all simple Lie algebras in key representations.

Findings

Explicit formulas for invariant projectors in defining and adjoint representations.

Construction of invariant solutions to the Yang-Baxter equations.

Universal description of simple Lie algebras using Vogel parameters.

Abstract

We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in defining (minimal fundamental) and adjoint representations. 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 two cases, when T is the defining and the adjoint representation. In the case when T is the defining representation, these projectors and the split Casimir operator are used to explicitly write down invariant solutions of the Yang-Baxter equations. 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.