The uniform structure of $\mathfrak{g}^{\otimes 4}$

TL;DR

This paper provides a universal decomposition of the fourth tensor power of the adjoint representation for all simple Lie algebras, including explicit formulas for dimensions and eigenvalues, advancing the understanding of Lie algebra representations.

Contribution

It introduces a uniform decomposition into Casimir eigenspaces for th powers of the adjoint representation with universal formulas, extending the theory of Lie algebra representations.

Findings

Decomposition into Casimir eigenspaces for th power of adjoint representation.

Universal formulas for dimensions and eigenvalues applicable to all simple Lie algebras.

Assumption of similar uniform decomposition for arbitrary powers of adjoint representations.

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, formulae 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 formulae exists for an arbitrary power of the adjoint representations.