Split Casimir Operator of Algebra $$D(2,1;\alpha )$$ in Representations $$a{{d}^{{ \otimes 2}}}$$ and $$a{{d}^{{ \otimes 3}}}$$ and Vogel Parameterization

Abstract

Characteristic identities for \(2\)- and \(3\)-split Casimir algebra operators \(D(2,1;\alpha )\) in representations \(a{{d}^{{ \otimes 2}}}\) and \(a{{d}^{{ \otimes 3}}}\) are found; projectors onto the invariant subspaces of these representations are constructed, and formulas for their supertraces are derived. All formulas agree with the universal description of subrepresentations of representations \(a{{d}^{{ \otimes 2}}}\) and \(a{{d}^{{ \otimes 3}}}\) of basic classical Lie superalgebras in terms of Vogel parameters.