RESEARCH ARTICLE

Batalin–Vilkovisky algebra structures on Hochschild cohomology of generalized Weyl algebras

  • Liyu LIU ,
  • Wen MA
Expand
  • School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China

Received date: 18 Sep 2020

Accepted date: 08 Sep 2021

Published date: 15 Oct 2022

Copyright

2022 Higher Education Press

Abstract

We devote to the calculation of Batalin–Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi–Yau generalized Weyl algebras. We first establish a Van den Bergh duality at the level of complex. Then based on the results of Solotar et al., we apply Kowalzig and Krähmer's method to the Hochschild homology of generalized Weyl algebras, and translate the homological information into cohomological one by virtue of the Van den Bergh duality, obtaining the desired Batalin–Vilkovisky algebra structures. Finally, we apply our results to quantum weighted projective lines and Podleś quantum spheres, and the Batalin–Vilkovisky algebra structures for them are described completely.

Cite this article

Liyu LIU , Wen MA . Batalin–Vilkovisky algebra structures on Hochschild cohomology of generalized Weyl algebras[J]. Frontiers of Mathematics in China, 2022 , 17(5) : 915 -941 . DOI: 10.1007/s11464-021-0978-6

1
Bavula V. Generalized Weyl algebras and their representations. Algebra i Analiz, 1992, 4: 75–97

2
Bavula V. Global dimension of generalized Weyl algebras. In: Bautista R, Martinez-Villa R, de la Pena J A, eds. Representation Theory of Aalgebras (ICRA VII, Cocoyoc, Mexico, August 22-26, 1994). CMS Conf Proc, Vol 18. 1996, 81–107

3
Bavula V. Tensor homological minimal algebras, global dimension of the tensor product of algebras and of generalized Weyl algebras. Bull Sci Math, 1996, 120: 293–335

4
Bavula V, Jordan D A. Isomorphism problems and groups of automorphisms for generalized Weyl algebras. Trans Amer Math Soc, 2001, 353: 769–794

DOI

5
Becchi C, Rouet A, Stora R. Renormalization of the Abelian Higgs–Kibble model. Comm Math Phys, 1975, 42: 127–162

DOI

6
Brzeziński T, Fairfax S A. Quantum teardrops. Comm Math Phys, 2012, 316: 151–170

DOI

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

DOI

8
Farinati M A, Solotar A, Suárez-Álvarez M. Hochschild homology and cohomology of generalized Weyl algebras. Ann Inst Fourier (Grenoble), 2003, 53: 465–488

DOI

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

DOI

10
Gerstenhaber M, Schack S D. Algebraic cohomology and deformation theory. In: Hazewinkel M, Gerstenhaber M, eds. Deformation Theory of Algebras and Structures and Applications. NATO Adv Sci Inst Ser C Math Phys Sci, Vol 247. Dordrecht: Kluwer Acad Publ, 1988, 11–264

DOI

11
Getzler E. Batalin–Vilkovisky algebras and two-dimensional topological field theories. Comm Math Phys, 1994, 159: 265–285

DOI

12
Ginzburg V. Calabi–Yau algebras. arXiv: 0612139

13
Huebschmann J. Lie–Rinehart algebras, Gerstenhaber algebras and Batalin–Vilkovisky algebras. Ann Inst Fourier (Grenoble), 1998, 48: 425–440

DOI

14
Kimura T, Voronov A, Stasheff J. On operad structures of moduli spaces and string theory. Comm Math Phys, 1995, 171: 1–25

DOI

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

DOI

16
Lambre T, Zhou G D, 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

DOI

17
Lian B H, Zukerman G J. New perspectives on the BRST-algebraic structure of string theory. Comm Math Phys, 1993, 154: 613–646

DOI

18
Liu L Y. Homological smoothness and deformations of generalized Weyl algebras. Israel J Math, 2015, 209: 949–992

DOI

19
Podleś P. Quantum spheres. Lett Math Phys, 1987, 14: 193–202

DOI

20
Solotar A, Suárez-Álvarez M, Vivas Q. Hochschild homology and cohomology of generalized Weyl algebras: the quantum case. Ann Inst Fourier (Grenoble), 2013, 63: 923–956

DOI

21
Tradler T. The Batalin–Vilkovisky algebra on Hochschild cohomology induced by infinity inner products. Ann Inst Fourier (Grenoble), 2008, 58: 2351–2379

DOI

22
Van den Bergh M. A relation between Hochschild homology and cohomology for Gorenstein rings. Proc Amer Math Soc, 1998, 126: 1345–1348. Erratum ibid. Proc Amer Math Soc, 2002, 130: 2809–2810

DOI

23
Xu P. Gerstenhaber algebras and BV-algebras in Poisson geometry. Comm Math Phys, 1999, 200: 545–560

DOI

Outlines

/