On universal quantum dimensions of certain two-parameter series of representations

Abstract

We present the universal, in Vogel's sense, expression for the quantum dimension of Cartan product of an arbitrary number of adjoint and $X_2$ representations of simple Lie algebras. The same formula mysteriously gives quantum dimensions of some other representations of the same Lie algebra under permutations of universal parameters. We list these representations for exceptional algebras and stable versions for classical algebras, when the rank of the classical algebra is sufficiently large w.r.t. the powers of representations. We show that universal formulae can have singularities on Vogel's plane for some algebras and that they give correct answers when restricted on appropriate lines on Vogel's plane. We note that the same irreducible representation can have several universal formulae for its (quantum) dimension, and discuss the implication of this phenomena on the Cohen - de Man method of calculation of universal formulae.