Uniqueness of universal quantum dimensions

We prove, under some assumptions, the uniqueness of universal quantum dimension formulae. We first show that the quantum dimensions of classical algebras have a specific form that can be represented universally, in Vogel’s sense, as the product/ratio of q-dimension factors with arguments $a\alpha +b\beta +c\gamma$ where $c=0,1,2$ and $a,b$ are integers. On this basis, we derive simple uniqueness criteria, according to which all known universal quantum dimensions are unique.