Hochschild homology of some quantum algebras
type: Item
instance of: scholarly article
title: Hochschild homology of some quantum algebras
zbMATH Open document ID: 0944.16007
DOI: 10.1016/S0022-4049(97)00102-3
published in: Journal of Pure and Applied Algebra
publication date: 2000-06-27T00:00:00Z
review text: Let \(k\) be a commutative ring and let \(n,r\) be positive integers with \(n\geq r\). The authors consider \(k\)-algebras, denoted \(S^r_{Q,P}\), generated by \(n\) indeterminates \(x_i\), \(1\leq i\leq n\), and the inverses of \(x_{r+1},\ldots x_n\), subject to relations of the form \(x_jx_i=q_{ij}x_ix_j+p_{ij}\), where the \(p_{ij}\)'s and \(q_{ij}\)'s are parameters from \(k\) satisfying \(q_{ii}=1\), \(q_{ji}q_{ij}=1\) for \(i<j\), and \(p_{ij}=0\) for \(j>r\). For such algebras they compute the Hochschild homology and cohomology, the method being to construct a homotopy equivalence between the canonical complex and a more tractible complex. They also study the Hochschild homology and cohomology of an Ore extension of the form \(S^r_{Q,P}[t;\alpha,\delta]\) where \(\alpha\) is a \(k\)-automorphism and \(\delta\) is an \(\alpha\)-derivation. Examples of such extensions for which explicit calculations are presented include the quantum enveloping algebra \(U_q(sl(2,k))\) and the coordinate ring of quantum \(2\times 2\) matrices over \(k\).
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1308249
zbMATH Keywords: Hochschild homology
zbMATH Keywords: Hochschild cohomology
zbMATH Keywords: quantum enveloping algebras
zbMATH Keywords: Ore extensions
zbMATH Keywords: coordinate rings of quantum matrices
author: Jorge A. Guccione
author: Juan J. Guccione
reviewed by: David A. Jordan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308249