Author’s summary: We analyse the noncommutative space underlying the quantum group \(\text{SU}_q(2)\) from the spectral point of view, which is the basis of non-commutative geometry, and show how the general theory developed in our joint work with H. Moscovici [Geom. Funct. Anal. 5, No. 2, 174–243 (1995; Zbl 0960.46048)] applies to the specific spectral triple defined by Chakraborty and Pal. This provides the pseudo-differential calculus, the Wodzciki-type residue, and the local cyclic co-cycle giving the index formula. The co-chain whose co-boundary is the difference between the original Chern character and the local one is given by the remainders in the rational approximation of the logarithmic derivative of the Dedekind eta function. This specific example allows us to illustrate the general notion of locality in non-commutative geometry. The formulae computing the residue are ‘local’. Locality by stripping all the expressions from irrelevant details makes them computable. The key feature of this spectral triple is its equivariance, i.e. the \(\text{SU}_q(2)\)-symmetry. We shall explain how this naturally leads to the general concept of invariant cyclic cohomology in the framework of quantum group symmetries.
In more detail, let \({\mathcal H}\) be a Hilbert space with orthonormal basis \(e^n_{ij}\); where \(n\in (1/2)N\) and \(i,j\in \{-n,-n+1,\dots, n\}\).
\[ \begin{aligned} \alpha e^n_{ij} &= a_+(n,i,j;q)e_{i-1/2,j-1/2}^{n+1/2} + a_-(n,i,j;q)e_{i-1/2,j-1/2}^{n-1/2},\\ \beta e_{ij}^n &= b_+(n,i,j; q)e_{i+1/2,j-1/2}^{n+1/2} + b_-(n,i,j; q)e_{i+1/2,j-1/2}^{n-1/2}.\\ f(\beta) &= \sum_{n>0} f_n\beta^n+ \sum_{n< 0} f_n(\beta^*)^{-n}+ f_0 e\text{ for }f= \sum f_n e^{in\theta}\in C^\infty(S^1).\end{aligned} \] \(A\) is the algebra given by \[ a= \sum_{k,m\geq 0}\alpha^{*k}f_{km}(\beta) \alpha^m+ \sum_{m\geq 0}\lambda_m \alpha^m+ \sum_{k< 0}\lambda_k \alpha^{*k},\qquad f_{km}\in C^\infty(S^1). \]
\(D:D(e_{ij}^n)= (2\delta_0(n-i)- 1)2ne^n_{ij}\), where \(\delta_0(k)= 0\) for \(k\neq 0\) and \(\delta_0(0)= 1\). The author’s works with Moscovich are applied to the above spectral triple \((A,{\mathcal H},D)\). Let \(\sigma(a)\in C^\infty(S^1)\) be the symbol of \(a\in A\), \(\partial=\partial_\beta- \partial_\alpha\), and \(da= \partial a+(1/2)\sigma(a)'' d\theta^2\).
Theorem: The local index formula of the triple \((A,{\mathcal H},D)\) for \(\text{SU}_q(2)\), \(q\in (0,1)\) is given by the cyclic cocycle \(\chi(a_0,a_1)= \int a_0 da_1\), \(\forall a_0,a_1\in A\), up to the coboundary of the cochain \((\phi_0,\phi_2)\). That is, the cocycle \(\psi_1= \chi\). The functions \(\phi_0((\beta^*\beta)^r)\) of the variable \(q\) are of the form \(q^{-2r}\left(q^2R_r(q^2)- G(q^2)\right)\), \(G(q^2)= q^2\partial_{q^2}\log\left(\eta(q^2)\right)\). \(\eta(q^2)\) is the Dedekind eta-function for \(q^2\), \(R_r\) the rational fraction.