Twisted Homology of Quantum SL(2) - Part II

Abstract

Abstract. We complete the calculation of the twisted cyclic homology of the quantised coordinate ring A = Cq[SL(2)] of SL(2) that we began in [14]. In particular, a nontrivial cyclic 3-cocycle is constructed which also has a nontrivial class in Hochschild cohomology and thus should be viewed as a noncommutative geometry analogue of a volume form.

A residue formula for the fundamental Hochschild class on the Podleś sphere

“…Analogous results for the spectral triple from [DLSSV] were obtained in [DLSSV2]. Contrasting these 'dimension drop' results, Hadfield and the first author [HK1,HK2] showed that SU q (2) is a twisted Calabi-Yau algebra of dimension 3 whose twist is the inverse of the modular automorphism for the Haar state on this compact quantum group, cf. Section 2.…”

“…In Section 2 we recall the definitions of SU q (2), the Haar state on SU q (2) and the associated GNS representation, and finally the modular theory of the Haar state. In Section 3 we recall the homological constructions of [HK1,HK2], and prove some elementary results we will need when we come to show that our residue cocycle does indeed recover the class of the fundamental cocycle. Section 4 contains all the key analytic results on meromorphic extensions of certain functions that allow us to prove novel summability type results for operators whose eigenvalues have mixed polynomial and exponential growth, see Lemma 4.2.…”

Twisted Cyclic Cohomology and Modular Fredholm Modules

“…It follows from results of Hadfield and Krähmer (see [11,Lemma 4.6]) that the map I : HH σ 3 (A(SU q (2))) → HC σ 3 (A(SU q (2))) is surjective. Therefore, the Hochschild cohomology class of the Chern character is nontrivial in HH 3 σ (A(SU q (2))).…”

Cyclic-Homology Chern–Weil Theory for Families of Principal Coactions

“…For instance, the fact that quasi-isomorphic chain complexes parameterized by a commutative base ring different from a field are not necessarily chain homotopy equivalent could be a potential way of more detailed analysis of the dimension-drop phenomena in deformation quantization [22]. Notice that, although in some cases the dimension drop can be prevented by the use of twisted Hochschild (and cyclic) homology [27][28][29][30], the Chern character always takes values in the untwisted cyclic homology.…”