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

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