Frobenius Poisson algebras

Abstract

This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology algebras. For any Frobenius Poisson algebra, all Batalin-Vilkovisky operators on its Poisson cochain complex are described explicitly. It is proved that there exists a Batalin-Vilkovisky operator on its cohomology algebra which is induced from a Batalin-Vilkovisky operator on the Poisson cochain complex, if and only if the Poisson structure is pseudo-unimodular. The relation between modular derivations of polynomial Poisson algebras and those of their truncated Poisson algebras is also described in some cases.

Keywords

Poisson algebra / Frobenius algebra / Batalin-Vilkovisky algebra / Poisson (co)homology / modular derivation

References

[1] Batalin I A, Vilkovisky G S. Quantization of gauge theories with linearly dependent generators. Phys Rev D, 1983, 28: 2567–2582

[2] Batalin I A, Vilkovisky G S. Closure of the gauge algebra, generalized Lie equations and Feynman rules. Nuclear Phys B, 1984, 234: 106–124

[3] Batalin I A, Vilkovisky G S. Existence theorem for gauge algebra. J Math Phys, 1985, 26: 172–184

[4] Berger R, Pichereau A. Calabi-Yau algebras viewed as deformations of Poisson algebras. Algebr Represent Theory, 2014, 17: 735–773

[5] Brylinski J L. A differential complex for Poisson manifolds. J Differential Geom, 1988, 28: 93–114

[6] Chen X, Chen Y, Eshmatov F, Yang S.Poisson cohomology, Koszul duality, and Batalin-Vilkovisky algebras. ArXiv: 1701.06112v2

[7] Chen X, Yang S, Zhou G.Batalin-Vilkovisky algebras and the noncommutative Poincaré duality of Koszul Calabi-Yau algebras. J Pure Appl Algebra, 2016, 220: 2500–2532

[8] Dolgushev V A. The Van den Bergh duality and the modular symmetry of a Poisson variety. Selecta Math (N S), 2009, 14: 199–228

[9] Gerstenhaber M. The cohomology structure of an associative ring. Ann of Math (2), 1963, 78: 267–288

[10] Ginzburg V.Lectures on Noncommutative Geometry. ArXiv: math.AG/0506603

[11] Huebschmann J. Poisson cohomology and quantization. J Reine Angew Math, 1990, 408: 57–113

[12] Huebschmann J. Duality for Lie-Rinehart algebras and the modular class. J Reine Angew Math, 1999, 510: 103–159

[13] Kontsevich M. Formality conjecture. In: Deformation Theory and Symplectic Geometry, Ascona, 1996. Math Phys Stud, Vol 20. Dordrecht: Kluwer Academic Publishers, 1997, 139–156

[14] Kontsevich M. Deformation quantization of Poisson manifolds. Lett Math Phys, 2003, 66: 157–216

[15] Kowalzig N, Krähmer U. Batalin-Vilkovisky structures on Ext and Tor. J Reine Angew Math, 2014, 697: 159–219

[16] Lambre T, Zhou G, Zimmermann A. The Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra. J Algebra, 2016, 446: 103–131

[17] Launois S, Richard L. Twisted Poincaré duality for some quadratic Poisson algebras. Lett Math Phys, 2007, 79: 161–174

[18] Launois S, Richard L. Poisson (co)homology of truncated polynomial algebras in two variables. Comptes Rendus Mathématique, 2009, 347: 133–138

[19] Laurent-Gengoux C, Pichereau A, Vanhaecke P.Poisson Structures. Grundlehren Math Wiss, Vol 347. Berlin: Springer, 2013

[20] Lichnerowicz A. Les varieties de Poisson et leurs algebres de Lie associees. J Differential Geom, 1977, 12: 253–300

[21] Luo J, Wang S Q, Wu Q S. Twisted Poincar? duality between Poisson homology and Poisson cohomology. J Algebra, 2015, 442: 484–505

[22] Luo J, Wang S Q, Wu Q S. Batalin-Vilkovisky structures and Poincaré dualities for smooth Poisson algebras. Preprint

[23] Marconnet N. Homologies of cubic Artin-Schelter regular algebras. J Algebra, 2004, 278: 638–665

[24] Maszczyk T. Maximal commutative subalgebras. Poisson geometry and Hochschild homology. ArXiv: math/0603386v1

[25] Oh S Q. Poisson enveloping algebras. Comm Algebra, 1999, 27: 2181–2186

[26] Shoikhet B. Koszul duality in deformation quantization and Tamarkin's approach to Kontsevich formality. Adv Math, 2010, 224: 731–771

[27] Tagne Pelap S R.On the Hochschild homology of elliptic Sklyanin algebras. Lett Math Phys, 2009, 87: 267–281

[28] Tagne Pelap S R. Poisson (co)homology of polynomial Poisson algebras in dimension four: Sklyanin's case. J Algebra, 2009, 322: 1151–1169

[29] Tradler T. The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products. Ann Inst Fourier (Grenoble), 2008, 58: 2351–2379

[30] Van den Bergh M. Noncommutative homology of some three-dimensional quantum spaces. K-Theory, 1994, 8: 213–230

[31] Wang S Q, Wu Q S, Zhou G D, Zhu C. Batalin-Vilkovisky structure over Frobenius Poisson cohomology. Preprint

[32] Weinstein A. Lecture on Symplectic Manifolds. CBMS Reg Conf Ser Math, No 29. Providence: Amer Math Soc, 1977

[33] Weinstein A. The local structure of Poisson manifolds. J Differential Geom, 1983, 18: 523–557

[34] Xu P. Gerstenhaber algebras and BV-algebras in Poisson geometry. Comm Math Phys, 1999, 200: 545–560

[35] Zhu C. Twisted Poincaré duality for Poisson homology and cohomology of affine Poisson algebras, Proc Amer Math Soc, 2015, 143: 1957–1967

[36] Zhu C, Van Oystaeyen F, Zhang Y H. On (co)homology of Frobenius Poisson algebras. J K-Theory, 2014, 14: 371–386

RIGHTS & PERMISSIONS

2019 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature