Split Casimir operator and universal formulation of the simple Lie algebras

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.

13 pages. arXiv admin note: substantial text overlap with arXiv:2102.08258

References in corpus (3)

• Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots
• On universal knot polynomials
• Diophantine equations, Platonic solids, McKay correspondence, equivelar maps and Vogel's universality