A residue formula for the fundamental Hochschild class on the Podleś sphere. (Q2874239)

This paper identifies the fundamental Hochschild cohomology class of the quantum algebra \(\mathcal{A}=\mathcal{O}\left( S_{q}^{2}\right) \) for the Podleś sphere \(S_{q}^{2}\) with \(q\in\left( 0,1\right) \) by a residue formula resembling that of Connes' pioneering work on noncommutative geometry in the framework of spectral triples.NEWLINENEWLINERecall that the Poincaré duality \(H_{2}\left( \mathcal{A},\, _{\sigma }\mathcal{A}\right) \cong H^{0}\left( \mathcal{A},\mathcal{A}\right) \cong\mathbb{C}\) is valid for the Hochschild (co)homology of the Podleś sphere, where \(\sigma\) is some automorphism of \(\mathcal{A}\) and the \(\mathcal{A}\)-bimodule \(_{\sigma}\mathcal{A}\) is the vector space \(\mathcal{A}\) with the two-sided \(\mathcal{A}\)-action given by \(a\vartriangleright b\vartriangleleft c:=\sigma\left( a\right) bc\) for \(a,b,c\in\mathcal{A}\). The cohomology class \(\left[ \phi\right] \in H^{2}\left( \mathcal{A},\left( _{\sigma}\mathcal{A}\right) ^{\ast}\right) \cong\left( H_{2}\left( \mathcal{A},\, _{\sigma}\mathcal{A}\right) \right) ^{\ast}\) of a nontrivial Hochschild 2-cocycle \(\phi\in\left( \mathcal{A} ^{\otimes3}\right) ^{\ast}\) on \(\mathcal{A}\) found earlier by Krähmer is naturally viewed as the fundamental Hochschild cohomology class of \(\mathcal{A}\). In this paper, the authors derive that \(\phi\left( a_{0} ,a_{1},a_{2}\right) \) equals the residue of tr\(\left( \gamma a_{0}\left[ D,a_{1}\right] \left[ D,a_{2}\right] K^{-2}\left| D\right| ^{-z}\right) \) at \(z=2\) in terms of the \(\mathcal{U}_{q}\left( \mathfrak{su}\left( 2\right) \right) \)-equivariant even spectral triple \(\left( \mathcal{A},\mathcal{H},D,\gamma\right) \) over \(\mathcal{A}\) constructed earlier by D{a}browski and Sitarz, where \(K\) is the standard group-like generator of \(\mathcal{U}_{q}\left( \mathfrak{su}\left( 2\right) \right) \).