A product-to-sum formula for the quantum group of SL(2,C)

Abstract

The paper exhibits a product-to-sum formula for the observables of a certain quantization of the moduli space of flat SU(2)-connections on the torus. This quantization was defined using the topological quantum field theory that was developed by Reshetikhin and Turaev from the quantum group of SL(2,C) at roots of unity. As a corollary it is shown that the algebra of quantum observables is a subalgebra of the noncommutative torus with rational rotation angle. The proof uses topological quantum field theory with corners, and is based on the description of the matrices of the observables in a canonical basis of the Hilbert space of the quantization.