On (ad)n(X2)k series of universal quantum dimensions

Abstract

We present a universal, in Vogel’s sense, expression for the quantum dimension of the Cartan product of arbitrary powers of the adjoint and X2 representations of simple Lie algebras. The same formula mysteriously yields quantum dimensions of some other representations of the same Lie algebra for permuted universal parameters, provided linear resolvability of singularities is applied. We list these representations for the exceptional algebras and their stable versions for the classical algebras (when the rank of the classical algebra is sufficiently large with regard to the powers of representations). Universal formulas may have singularities at the points in Vogel’s plane, corresponding to some simple Lie algebras. We prove that our formula is linearly resolvable at all those singular points, i.e., yields finite answers when restricted either on the classical or the exceptional lines, and make a conjecture that these answers coincide with (quantum) dimensions of some irreducible representations. In a number of cases, particularly, in the case of so(8) algebra, which belongs both to the orthogonal and the exceptional lines, it is confirmed that both resolutions yield relevant answers. We note that an irreducible representation may have several universal formulas for its (quantum) dimension and discuss the impact of this phenomenon to the method presented by Cohen and de Man [C. R.Acad. Sci., Ser. 1 322(5), 427–432 (1996)] for derivation of universal formulas.