\(3\)-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

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.