A note on the differential calculus of Hochschild theory for $ A_{\infty} $-algebras

Youming Chen; Weiguo Lyu; Song Yang; Youming Chen; Weiguo Lyu; Song Yang

doi:10.3934/era.2022163

Electronic Research Archive

2022, Volume 30, Issue 9: 3211-3237. doi: 10.3934/era.2022163

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Research article

A note on the differential calculus of Hochschild theory for $ A_{\infty} $-algebras

1.
School of Science, Chongqing University of Technology, 69 Hongguang Avenue, Chongqing, MO 400054, China
2.
School of Mathematical Sciences, Anhui University, 111 Jiulong Road, Hefei, MO 230039, China
3.
Center for Applied Mathematics, Tianjin University, 92 Weijin Road, Tianjin, MO 300072, China

Received: 28 June 2020 Revised: 21 May 2022 Accepted: 31 May 2022 Published: 20 June 2022

We show by constructing explicit homotopy operators that the Hochschild (co)homology of an $ A_{\infty} $-algebra of Stasheff admits a differential calculus structure. As an application, we reproduce a result of Tradler which says that the Hochschild cohomology of a cyclic $ A_\infty $-algebra admits a Batalin-Vilkovisky algebra structure.
- differential calculus,
- $ A_{\infty} $-algebra,
- cyclic $ A_{\infty} $-algebra,
- Hochschild (co)homology,
- Batalin-Vilkovisky algebra
Citation: Youming Chen, Weiguo Lyu, Song Yang. A note on the differential calculus of Hochschild theory for $ A_{\infty} $-algebras[J]. Electronic Research Archive, 2022, 30(9): 3211-3237. doi: 10.3934/era.2022163

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Abstract

We show by constructing explicit homotopy operators that the Hochschild (co)homology of an $ A_{\infty} $-algebra of Stasheff admits a differential calculus structure. As an application, we reproduce a result of Tradler which says that the Hochschild cohomology of a cyclic $ A_\infty $-algebra admits a Batalin-Vilkovisky algebra structure.

References

[1]	S. Kobayashi, K. Nomizu, Foundations of differential geometry, 1. Interscience Publishers, a Division of John Wiley & Sons, New York-London, 1963.
[2]	Y. L. Daletskii, I. M. Gelfand, B. L. Tsygan, On a variant of noncommutative differential geometry, Soviet Math. Dokl., 40 (1990), 422–426.
[3]	D. Tamarkin, B. Tsygan, The ring of differential operators on forms in noncommutative calculus, Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 73 (2005), 105–131. https://doi.org/10.1090/pspum/073/2131013
[4]	V. A. Dolgushev, D. E. Tamarkin, B. L. Tsygan, Noncommutative calculus and the Gauss-Manin connection, Higher structures in geometry and physics, Progr. Math., 287, Birkhäuser/Springer, New York, (2011), 139–158. https://doi.org/10.1007/978-0-8176-4735-3_7
[5]	M. Kontsevich, Y. Soibelman, Notes on $A_\infty$-algebras, $A_\infty$-categories and non-commutative geometry. I, Homological mirror symmetry, 153-219, Lecture Notes in Phys., 757, Springer, Berlin, 2009. https://doi.org/10.1007/978-3-540-68030-7_6
[6]	V. Ginzburg, Calabi-Yau algebras, arXiv preprint math/0612139, 2006.
[7]	L. de Thanhoffer de Völcsey, M. Van den Bergh, Calabi-Yau deformations and negative cyclic homology, J. Noncommut. Geom., 12 (2018), 1255–1291. https://doi.org/10.4171/JNCG/304 doi: 10.4171/JNCG/304
[8]	S. B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc., 152 (1970), 39–60. https://doi.org/10.1090/S0002-9947-1970-0265437-8 doi: 10.1090/S0002-9947-1970-0265437-8
[9]	T. Tradler, The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products, Ann. Inst. Fourier (Grenoble), 58 (2008), 2351–2379. https://doi.org/10.5802/aif.2417 doi: 10.5802/aif.2417
[10]	X. Chen, S. Yang, G. Zhou, Batalin-Vilkovisky algebras and the noncommutative Poincaré duality of Koszul Calabi-Yau algebras, J. Pure Appl. Algebra, 220 (2016), 2500–2532. https://doi.org/10.1016/j.jpaa.2015.11.016 doi: 10.1016/j.jpaa.2015.11.016
[11]	R. Berger, Koszulity for nonquadratic algebras, J. Algebra, 239 (2001), 705–734. https://doi.org/10.1006/jabr.2000.8703 doi: 10.1006/jabr.2000.8703
[12]	M. Van den Bergh, Calabi-Yau algebras and superpotentials, Sel. Math. New Ser., 21 (2015), 555–603. https://doi.org/10.1007/s00029-014-0166-6 doi: 10.1007/s00029-014-0166-6
[13]	L. Menichi, Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras, K-Theory, 3 (2004), 231–251. https://doi.org/10.1007/s10977-004-0480-4 doi: 10.1007/s10977-004-0480-4
[14]	M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math., 78 (1963), 267–288. https://doi.org/10.2307/1970343 doi: 10.2307/1970343
[15]	G. Hochschild, On the cohomology groups of an associative algebra, Ann. Math., 46 (1945), 58–67. https://doi.org/10.2307/1969145 doi: 10.2307/1969145
[16]	L. Menichi, Batalin-Vilkovisky algebra structures on Hochschild cohomology, Bull. Soc. Math. France, 137 (2009), 277–295. https://doi.org/10.24033/bsmf.2576 doi: 10.24033/bsmf.2576
[17]	T. Lambre, Dualité de Van den Bergh et Structure de Batalin-Vilkovisky sur les algèbres de Calabi-Yau, J. Noncommut. Geom., 4 (2010), 441–457. https://doi.org/10.4171/JNCG/62 doi: 10.4171/JNCG/62
[18]	N. Kowalzig, U. Krähmer, Batalin-Vilkovisky structures on Ext and Tor, J. Reine Angew. Math., 697 (2014), 159–219. https://doi.org/10.1515/crelle-2012-0086 doi: 10.1515/crelle-2012-0086
[19]	T. Lambre, G. Zhou, A. Zimmermann, The Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra, J. Algebra, 446 (2016), 103–131. https://doi.org/10.1016/j.jalgebra.2015.09.018 doi: 10.1016/j.jalgebra.2015.09.018
[20]	J. D. Stasheff, Homotopy associativity of H-spaces, Ⅱ, Trans. Amer. Math. Soc., 108 (1963), 293–312. https://doi.org/10.1090/S0002-9947-1963-0158400-5 doi: 10.1090/S0002-9947-1963-0158400-5
[21]	M. Gerstenhaber, A. A. Voronov, Homotopy G-Algebras and Moduli Space Operad, Int. Math. Res. Not., 3 (1995), 141–153. https://doi.org/10.1155/S1073792895000110 doi: 10.1155/S1073792895000110
[22]	E. Getzler, Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology, Israel Math. Conf. Proc., 7 (1993), 65–78.
[23]	E. Getzler, J. D. S. Jones, $A_{\infty}$-algebras and the cyclic bar complex, Illinois J. Math., 34 (1990), 256–283. https://doi.org/10.1215/ijm/1255988267 doi: 10.1215/ijm/1255988267
[24]	E. Getzler, J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, arXiv preprint, hep-th/9403055, 1994.
[25]	B. L. Tsygan, Noncommutative Calculus and Operads, Topics in Noncommutative Geometry, 16 (2012), 19–66.

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