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By Michel Van, Den Bergh and Communicated Lance W. Small


Abstract. Let “HH ” stand for Hochschild (co)homology. In this note we show that for many rings A there exists d ∈ N such that for an arbitrary A-bimodule N we have HHi (N)=HHd−i(N). Such a result may be viewed as an analog of Poincaré duality. Combining this equality with a computation of Soergel allows one to compute the Hochschild homology of a regular minimal primitive quotient of an enveloping algebra of a semisimple Lie algebra, answering a question of Polo. In the sequel the base field will be denoted by k. Letgbeasemisimple Lie algebra and let A be a regular minimal primitive quotient of U(g). The Hochschild cohomology of A was computed by Soergel in [6] and shown to be equal to the cohomology of the corresponding flag variety. Soergel’s computation is rather ingenious and makes use of the Bernstein-Beilinson theorem together with the Riemann-Hilbert correspondence. The case of singular A is still open. After Soergel’s result Patrick Polo asked whether perhaps the Hochschild homology of A also coincided with the homology of the underlying flag manifold. It i

Year: 2013
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