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

TL;DR

This paper derives explicit formulas for projectors onto invariant subspaces of the ad^{⊗2} representation of certain Lie algebras, using the split Casimir operator and Vogel parametrization, enhancing understanding of their structure.

Contribution

It provides new explicit formulas for invariant projectors in the ad^{⊗2} representation of so(N) and sp(2r), connecting them with Vogel's universal Lie algebra framework.

Findings

Explicit projectors derived for so(N) and sp(2r)

Connection established with Vogel parametrization

Enhanced understanding of invariant subspace structure

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.