Non-commutative integration, zeta functions and the Haar state for $SU_{q}(2)$

Abstract

We study a notion of non-commutative integration, in the spirit of modular spectral triples, for the quantum group $SU_{q}(2)$. In particular we define the non-commutative integral as the residue at the spectral dimension of a zeta function, which is constructed using a Dirac operator and a weight. We consider the Dirac operator introduced by Kaad and Senior and a family of weights depending on two parameters, which are related to the diagonal automorphisms of $SU_{q}(2)$. We show that, after fixing one of the parameters, the non-commutative integral coincides with the Haar state of $SU_{q}(2)$. Moreover we can impose an additional condition on the zeta function, which also fixes the second parameter. For this unique choice the spectral dimension coincides with the classical dimension.