The Van den Bergh duality and the modular symmetry of a Poisson variety

Abstract

We consider a smooth Poisson affine variety with the trivial canonical bundle over complex numbers. For such a variety the deformation quantization algebra A_h enjoys the conditions of the Van den Bergh duality theorem and the corresponding dualizing module is determined by an outer automorphism of A_h intrinsic to A_h. We show how this automorphism can be expressed in terms of the modular class of the corresponding Poisson variety. We also prove that the Van den Bergh dualizing module of the deformation quantization algebra A_h is free if and only if the corresponding Poisson structure is unimodular.

arXiv comment28 pages

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Cited by in corpus (6)

• On the Heisenberg invariance and the Elliptic Poisson tensors
• Holomorphic Koszul-Brylinski Homology
• On the Hochschild (co)homology of Quantum Homogeneous Spaces
• Duality and products in algebraic (co)homology theories
• Modular classes of Lie algebroid morphisms
• Integer solutions of integral inequalities and H-invariant Jacobian Poisson structures