Projectors on invariant subspaces of representations $$\mathrm{ad}^{\otimes2}$$ of Lie algebras $$so(N)$$ and $$sp(2r)$$ and Vogel parameterization

Abstract

Using the split Casimir operator, we find explicit formulas for the projectors onto invariant subspaces of the $$ \mathrm{ad} ^{ \otimes 2}$$ representation of the algebras $$so(N)$$ and $$sp(2r)$$ . We also consider these projectors from the standpoint of the universal description of complex simple Lie algebras using the Vogel parameterization.

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