The paper is organized as follows. In Section 1 we recall basic facts about modular tensor categories (MTC), in particular, the action of modular group and various symmetries of this action. In Section 2 we define an inner product on the space of intertwiners in modular tensor categories with some additional properties (Hermitian MTC’s), and prove that the action of modular group is unitary with respect to this linear product.
In Section 3 we recall, following H. H. Andersen [Commun. Math. Phys. 149, No. 1, 149-159 (1992; Zbl 0760.17004)], the construction of MTC from representations of quantum groups of roots of unity. In Section 4 we show that this category can be endowed with a natural Hermitian structure.
Section 5 is devoted to a special case of the constructions above; namely, we let \({\mathfrak g}={\mathfrak s}{\mathfrak l}_n\) and take \(U\) to be a symmetric power of fundamental representation. We show that in this case the \(S\)-matrix can be written in terms of values of Macdonald’s polynomials of type \(A_{n-1}\) at roots of unity, which gives many identities for these special values. These expressions coincide with Cherednik’s formulas for difference Fourier transform.
Sections 6 and 7 are devoted to further study of MTC’s coming from quantum groups at roots of unity. In particular, we describe the Grothendieck ring of these categories (which is not new); we also give another description of the Hermitian structure on them.
In subsequent papers we will apply the same construction to the modular tensor category arising from the affine Lie algebras.