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

Abstract

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.