On the Hochschild homology of quantum \(\text{SL}(N)\) (Q2499717)

It is well-known that the Hochschild homology of quantized coordinate rings is not as well-behaved as for classical coordinate rings. In earlier work [\(K\)-Theory 34, No. 4, 327--360 (2005; Zbl 1098.58006)], the authors showed that in the case of \(A=k_q[\text{SL}(2)]\) this may be overcome by passing to Hochschild homology with coefficients in a suitable bimodule, \(A_\sigma\), yielding \(H_3(A,A_\sigma)\simeq k\). This result is extended here, where it is shown that for \(A=k_q[\text{SL}(N)]\), \(H_{N^2-1}(A,A_\sigma)\simeq k\). The bimodule \(A_\sigma\) is \(A\) with bimodule structure twisted by the modular automorphism \(\sigma\), this being the unique automorphism such that \(h(xy)=h(\sigma(y)x)\) where \(h\colon A\to k\) is the Haar functional. The proof of the above isomorphism makes use of an analogue of Poincaré duality for Hochschild (co)homology given by \textit{M. Van den Bergh} [Proc. Am. Math. Soc. 126, No. 5, 1345--1348 (1998); erratum ibid. 130, No. 9, 2809--2810 (2002; Zbl 0894.16005)].