Universality of dual quantum dimensions

Determine whether the dual quantum dimensions of simple Lie algebras can be universalized in Vogel parameters alongside the ordinary quantum dimensions, including universal formulas that coincide for simply laced root systems and correctly distinguish the non-simply-laced cases.

Background

The paper defines dual quantum dimensions by evaluating irreducible Lie-algebra characters at the dual Weyl vector rather than at the ordinary Weyl vector. For simply laced root systems, the two Weyl vectors coincide, so ordinary and dual quantum dimensions agree; for non-simply-laced systems, they generally differ.

The authors identify substantial obstructions to a universal Vogel-parameter formula. Any such construction would need to provide formulas agreeing on simply laced families while differing on non-simply-laced families, despite the geometry of Vogel’s parameter space. In addition, the dual quantum dimensions for the B_n and C_n families coincide in the adjoint representation even though those families correspond to distinct Vogel parameters and have different rank dependence. The paper does not resolve whether a satisfactory universalization exists.