Projectors on invariant subspaces of representations $\operatorname{ad}^{\otimes 2}$ of Lie algebras $so(N)$ and $sp(2r)$ and Vogel parametrization

Abstract

Explicit formulae for the projectors onto invariant subspaces of the $\operatorname{ad}^{\otimes 2}$ representation of the Lie algebras $so(N)$ and $sp(2r)$ have been found by means of the split Casimir operator. These projectors have also been considered from the viewpoint of the universal complex simple Lie algebra description by using the Vogel parametrisation.