Homological unimodularity and Calabi–Yau condition for Poisson algebras

Homological unimodularity and Calabi–Yau condition for Poisson algebras In this paper, we show that the twisted Poincaré duality between Poisson homology and cohomology can be derived from the Serre invertible bimodule. This gives another definition of a unimodular Poisson algebra in terms of its Poisson Picard group. We also achieve twisted Poincaré duality for Hochschild (co)homology of Poisson bimodules using rigid dualizing complex. For a smooth Poisson affine variety with the trivial canonical bundle, we prove that its enveloping algebra is a Calabi–Yau algebra if the Poisson structure is unimodular. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Letters in Mathematical Physics Springer Journals

Homological unimodularity and Calabi–Yau condition for Poisson algebras

Homological unimodularity and Calabi–Yau condition for Poisson algebras

Lett Math Phys (2017) 107:1715–1740 DOI 10.1007/s11005-017-0967-6 Homological unimodularity and Calabi–Yau condition for Poisson algebras 1 2 Jiafeng Lü · Xingting Wang · Guangbin Zhuang Received: 31 October 2016 / Revised: 7 April 2017 / Accepted: 19 April 2017 / Published online: 24 May 2017 © Springer Science+Business Media Dordrecht 2017 Abstract In this paper, we show that the twisted Poincaré duality between Poisson homology and cohomology can be derived from the Serre invertible bimodule. This gives another definition of a unimodular Poisson algebra in terms of its Poisson Picard group. We also achieve twisted Poincaré duality for Hochschild (co)homology of Poisson bimodules using rigid dualizing complex. For a smooth Poisson affine variety with the trivial canonical bundle, we prove that its enveloping algebra is a Calabi–Yau algebra if the Poisson structure is unimodular. Keywords Poisson algebra · Calabi–Yau algebra · Hochschild (co)homology · Poisson (co)homology · Dualizing complex Mathematics Subject Classification 16E40 · 17B35 · 17B63 B Jiafeng Lü jiafenglv@zjnu.edu.cn ; jiafenglv@gmail.com Xingting Wang xingting@temple.edu Guangbin Zhuang gbzhuang1981@gmail.com Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, People’s Republic of China Department of Mathematics, Temple University, Philadelphia, PA 19122, USA Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA 123 1716 J. Lü et al. 0 Introduction Poisson geometry is originated in classical mechanics where one describes the time evolution of a mechanical system by solving Hamiltons equations in terms of the Hamiltonian vector field. This inspires the definition of a Poisson manifold M which is equipped with a Lie bracket (called Poisson bracket) on the algebra C (M ) of smooth functions on M subject to...
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Publisher
Springer Netherlands
Copyright
Copyright © 2017 by Springer Science+Business Media Dordrecht
Subject
Physics; Theoretical, Mathematical and Computational Physics; Complex Systems; Geometry; Group Theory and Generalizations
ISSN
0377-9017
eISSN
1573-0530
D.O.I.
10.1007/s11005-017-0967-6
Publisher site
See Article on Publisher Site

Abstract

In this paper, we show that the twisted Poincaré duality between Poisson homology and cohomology can be derived from the Serre invertible bimodule. This gives another definition of a unimodular Poisson algebra in terms of its Poisson Picard group. We also achieve twisted Poincaré duality for Hochschild (co)homology of Poisson bimodules using rigid dualizing complex. For a smooth Poisson affine variety with the trivial canonical bundle, we prove that its enveloping algebra is a Calabi–Yau algebra if the Poisson structure is unimodular.

Journal

Letters in Mathematical PhysicsSpringer Journals

Published: May 24, 2017

References

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