The Uniform Structure of \(\mathfrak{g}^{\otimes 4}\)

Abstract

We obtain a uniform decomposition into Casimir eigenspaces (most of which are irreducible) of the fourth power of the adjoint representation \(\mathfrak{g}^{\otimes 4}\) for all simple Lie algebras. We present universal, in Vogel’s sense, formulas for the dimensions and split Casimir operator’s eigenvalues of all terms in this decomposition. We assume that a similar uniform decomposition into Casimir eigenspaces with universal dimension formulas exists for an arbitrary power of the adjoint representations.

DOI 10.1134/S1061920824030038