Frontiers of Mathematics in China >
Batalin–Vilkovisky algebra structures on Hochschild cohomology of generalized Weyl algebras
Received date: 18 Sep 2020
Accepted date: 08 Sep 2021
Published date: 15 Oct 2022
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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.
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
|
5 |
Becchi C, Rouet A, Stora R. Renormalization of the Abelian Higgs–Kibble model. Comm Math Phys, 1975, 42: 127–162
|
6 |
Brzeziński T, Fairfax S A. Quantum teardrops. Comm Math Phys, 2012, 316: 151–170
|
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
|
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
|
9 |
Gerstenhaber M. The cohomology structure of an associative ring. Ann of Math, 1963, 78: 267–288
|
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
|
11 |
Getzler E. Batalin–Vilkovisky algebras and two-dimensional topological field theories. Comm Math Phys, 1994, 159: 265–285
|
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
|
14 |
Kimura T, Voronov A, Stasheff J. On operad structures of moduli spaces and string theory. Comm Math Phys, 1995, 171: 1–25
|
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 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
|
17 |
Lian B H, Zukerman G J. New perspectives on the BRST-algebraic structure of string theory. Comm Math Phys, 1993, 154: 613–646
|
18 |
Liu L Y. Homological smoothness and deformations of generalized Weyl algebras. Israel J Math, 2015, 209: 949–992
|
19 |
Podleś P. Quantum spheres. Lett Math Phys, 1987, 14: 193–202
|
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
|
21 |
Tradler T. The Batalin–Vilkovisky algebra on Hochschild cohomology induced by infinity inner products. Ann Inst Fourier (Grenoble), 2008, 58: 2351–2379
|
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
|
23 |
Xu P. Gerstenhaber algebras and BV-algebras in Poisson geometry. Comm Math Phys, 1999, 200: 545–560
|
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