A. P. Isaev, A. A. Provorov, “$3$-split Casimir operator of the $sl(M|N)$ and $osp(M|N)$ simple Lie superalgebras in the representation $\operatorname{ad}^{\otimes 3}$ and the Vogel parameterization”, TMF, 2024, Volume 221, Number 1,Pages 154

$3$-split Casimir operator of the $sl(M|N)$ and $osp(M|N)$ simple Lie superalgebras in the representation $\operatorname{ad}^{\otimes 3}$ and the Vogel parameterization

A. P. Isaev^abc, A. A. Provorov^ac

^a Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia
^b Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia
^c St. Petersburg Department of the Steklov Mathematical Institute, Russian Academy of Sciences, St. Petersburg, Russia

Abstract: We find universal characteristic identities for the $3$-split casimir operator in the representation $\operatorname{ad}^{\otimes 3}$ of the $osp(m|n)$ and $sl(m|n)$ lie superalgebras. Using these identities, we construct projectors onto the invariant subspaces of these representations and find universal formulas for their superdimensions. All the formulas are in accordance with the universal description of subrepresentations of the $\operatorname{ad}^{\otimes 3}$ representation of simple basic Lie superalgebras in terms of the Vogel parameters.

Keywords: invariant subspace, projector, simple Lie superalgebra, split Casimir operator, Vogel parameters.

Received: 24.04.2024
Revised: 14.06.2024

DOI: 10.4213/tmf10744