Universal decomposition of k-th power of the adjoint into Casimir eigenspaces

Establish that for any integer k ≥ 1, the k-th tensor power of the adjoint representation of a simple Lie algebra admits a universal decomposition into Casimir eigenspaces within Vogel’s universal Lie algebra framework.

Background

The paper proves that for stable sequences of su(N) representations appearing in the decomposition of ad(su(N))^⊗k, the Casimir eigenvalues are linear in N. This linearity implies that these eigenvalues can be written in Vogel’s universal form, with parameters (α,β,γ)=(-2,2,N) for su(N).

By connecting this linearity to universal expressions for Casimir eigenvalues, the result provides evidence toward a broader claim proposed in prior work: that the k-th power of the adjoint representation admits a universal decomposition into Casimir eigenspaces across simple Lie algebras. The authors explicitly reference this as a hypothesis that their results support, indicating it remains an unproven claim.