The split Casimir operator and solutions of the Yang-Baxter equation for the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras, higher Casimir operators, and the Vogel parameters

TL;DR

This paper derives characteristic identities and invariant projectors for the split Casimir operator in $osp(M|N)$ and $s ext{l}(M|N)$ superalgebras, constructing solutions to the Yang-Baxter equation and higher Casimir operators using Vogel parameters.

Contribution

It introduces new characteristic identities and invariant projectors for the split Casimir operator in superalgebras, and constructs universal solutions to the Yang-Baxter equation.

Findings

Invariant projectors onto subspaces of tensor products are constructed.

Yang-Baxter solutions are expressed as rational functions of the split Casimir.

A universal generating function for higher Casimir operators is developed.

Abstract

We find the characteristic identities for the split Casimir operator in the defining and adjoint representations of the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras. These identities are used to build the projectors onto invariant subspaces of the representation $T^{\otimes 2}$ of the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras in the cases when $T$ is the defining and adjoint representations. For defining representations, the $osp(M|N)$- and $s\ell(M|N)$-invariant solutions of the Yang-Baxter equation are expressed as rational functions of the split Casimir operator. For the adjoint representation, the characteristic identities and invariant projectors obtained are considered from the viewpoint of a universal description of Lie superalgebras by means of the Vogel parametrization. We also construct a universal generating function for higher Casimir operators of the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras in the adjoint representation.