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

arXiv commentMinor corrections, 45 pages

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• Macdonald deformation of Vogel's universality and link hyperpolynomials
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• On Refined Vogel's universality
• Vogel's universality and Macdonald dimensions
• Torus knots in adjoint representation and Vogel's universality
• Vogel's universality and the classification problem for Jacobi identities
• Tau-functions beyond the group elements