Hochschild and cyclic homology of Ore extensions and some examples of quantum algebras (Q1372608)

The goal of this paper is the computation of the Hochschild and cyclic homology of Ore extensions \(A[t,\alpha,\delta]\) in the case that \(\alpha\) is an isomorphism. The authors introduce an acyclic double chain complex for \(A[t,\alpha,\delta]\) whose total complex is quasi-isomorphic to the normalized Hochschild complex. Then they study a natural filtration of this complex in order to reduce (under certain assumptions on \(A\)) the Hochschild (resp. cyclic) homology of \(A[t,\alpha,\delta]\) to the Hochschild (resp. cyclic) homology of \(A\). This is applied to compute (under a certain assumption on the parameters) the Hochschild and cyclic homology of the multiparameter quantum affine space which was already obtained by \textit{M. Wambst} [Ann. Inst. Fourier 43, No. 4, 1089-1156 (1993; Zbl 0810.16010)]. The authors also provide a reduction formula for the Hochschild homology of certain algebras of differential operators on the multiparameter quantum affine space in the ``generic'' case. If \(A\) is commutative and satisfies a strong regularity condition, they simplify the acyclic double complex further and thereby show that the Hochschild (resp. cyclic) homology of \(A[t,\alpha,\delta]\) and the Hochschild (resp. cyclic) homology of \(A\) coincide if no parameter is a root of unity. In particular, this applies to the algebra of \(q\)-difference operators where the corresponding complex was constructed by \textit{C. Kassel} [Commun. Math. Phys. 146, No. 2, 343-356 (1992; Zbl 0761.17020)].