Hochschild and cyclic homology of \(q\)-difference operators (Q1912819)

Denoting by \(k\) an arbitrary field, let \(D_q\) \((q\in k\setminus \{0\})\) be the \(k\)-algebra generated by \(x\), \(x^{-1}\), \(\partial\) and the relation \(\partial x-qx=1\), i.e.the \(q\)-analogue of \(D_1\), the \(k\)-algebra generated by \(x\), \(x^{-1}\), \(\partial\) and the Heisenberg relation \(\partial x- x\partial=1\), \textit{C. Kassel} [Commun. Math. Phys. 146, No. 2, 343-356 (1992; Zbl 0761.17020)] studied this algebra and shows that the Hochschild homology of \(D_q\) is the homology of a complex \(\mathbb{R}_{*,*}(D_q)\) simpler than the canonical one of Hochschild. In this paper, following the results of Kassel, the authors compute the Hochschild and cyclic homologies of \(D_q\) where \(q\) is a primitive \(m\)-th root of unity with \(m>1\) (when \(q=1\) or \(q^n\neq 1\) for all \(n\in\mathbb{N}\) these cohomologies are studied by Kassel in the previous paper, by \textit{W. Wodzicki} [Duke Math. J. 54, 641-647 (1987; Zbl 0635.18010)] and by the authors of the present paper in the preprint: Hochschild and cyclic homology of Ore's extensions and some quantum examples. So, in the first part, the Hochschild homology of \(D_Q\) is computed by means of the study of the natural filtration of \(\mathbb{R}_{**}(D_q)\). In the second part, an explicit formula for the morphisms \(B_*: HH_0(D_q)\to HH_1(D_q)\) and \(B_*: HH_1(D_q)\to HH_2(D_q)\) [see \textit{J. L. Loday}, Cyclic homology, Berlin, Springer (1992; Zbl 0780.18009)] is given. This fact together with the Gysin-Connes exact sequence permits to the authors to compute cyclic homology of \(D_q\).