Hochschild homology of some quantum algebras
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 \(k\).
scientific article; zbMATH DE number 1308249