On the Hochschild homology of quantum SL(N)

TL;DR

This paper proves a duality property for Hochschild homology of the quantum coordinate ring of SL(N), extending previous results from SL(2) to general N, revealing deep algebraic symmetries.

Contribution

It demonstrates that the quantum SL(N) coordinate ring satisfies a Poincare duality analogue for Hochschild (co)homology with a specific dualizing bimodule, generalizing prior SL(2) results.

Findings

Hochschild homology satisfies Poincare duality for quantum SL(N).

The top Hochschild homology group is one-dimensional.

The results extend previous work from SL(2) to general N.

Abstract

We show that the standard quantized coordinate ring A of quantum SL(N) satisfies van den Bergh's analogue of Poincare duality for Hochschild (co)homology with dualizing bimodule being A_sigma, the A-bimodule which is A as k-vector space with right multiplication twisted by the modular automorphism sigma of the Haar functional. This implies that H_{N^2-1} (A, A_sigma)=k, generalizing our previous results for quantum SL(2).