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

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