×

Homotopy Gerstenhaber algebras are strongly homotopy commutative. (English) Zbl 1469.16023

Homotopy Gerstenhaber algebras were introduced by A. A. Voronov and M. Gerstenhaber [Funct. Anal. Appl. 29, No. 1, 1–5 (1995; Zbl 0849.16010); translation from Funkts. Anal. Prilozh. 29, No. 1, 1–6 (1995)]. Singular cochain complexes are examples of homotopy Gerstenhaber algebras besides the Hochschild cochains of associative algebras. According to J. Stasheff and S. Halperin [in: Proc. adv. Study Inst. algebraic Topol.various Publ. Ser. 13, 567–577 (1970; Zbl 0224.55027)], the DG algebra is a strongly homotopy commutative algebra \(A\) if the multiplication \(\mu_{A}:A\otimes A\to A\) extends to an \(A_{\infty}\) morphism \(\Phi:A\otimes A\to A\) in the sense that the base component \(\Phi_{(1)}:A\otimes A\to A\) of \(\Phi\) equals \(\mu\). H. J. Munkholm [J. Pure Appl. Algebra 5, 1–50 (1974; Zbl 0294.55011)] alternatively defined it under \(3\) additional conditions. The author shows that any homotopy Gerstenhaber algebra is naturally a strongly homotopy commutative algebra in the sense of Stasheff-Halperin with a homotopy associative structure map. In the presence of certain additional operations corresponding to a \(\cup_1\)-product on the bar construction, the author shows that the structure map becomes homotopy commutative, which means that it is an strongly homotopy commutative algebra in the sense of Munkholm.

MSC:

16E45 Differential graded algebras and applications (associative algebraic aspects)
57T30 Bar and cobar constructions

Software:

SymPy
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baues, H-J, The cobar construction as a Hopf algebra, Invent. Math., 132, 467-489 (1998) · Zbl 0912.55015
[2] Belmans, P.: Hochschild (co)homology, and the Hochschild-Kostant-Rosenberg decomposition, course notes (2018). http://pbelmans.ncag.info/teaching/hh-2018
[3] Berger, C.; Fresse, B., Combinatorial operad actions on cochains, Math. Proc. Camb. Philos. Soc., 137, 135-174 (2004) · Zbl 1056.55006
[4] Franz, M.: The cohomology rings of homogeneous spaces, arXiv:1907.04777v2
[5] Franz, M.: Szczarba’s twisting cochain is comultiplicative, arXiv:2008.08943v1
[6] Gugenheim, VKAM; Munkholm, HJ, On the extended functoriality of \(\operatorname{Tor}\) and \(\operatorname{Cotor} \), J. Pure Appl. Algebra, 4, 9-29 (1974) · Zbl 0358.18015
[7] Hess, K.; Parent, P-E; Scott, J.; Tonks, A., A canonical enriched Adams-Hilton model for simplicial sets, Adv. Math., 207, 847-875 (2006) · Zbl 1112.55010
[8] Kadeishvili, T.: Cochain operations defining Steenrod \(\smile_i\)-products in the bar construction, Georgian Math. J. 10 (2003), 115-125. http://www.emis.de/journals/GMJ/vol10/v10n1-9.pdf · Zbl 1056.55007
[9] Kadeishvili, T., Measuring the noncommutativity of DG-algebras, J. Math. Sci. (N.Y.), 119, 494-512 (2004) · Zbl 1082.57029
[10] McClure, JE; Smith, JH, Multivariable cochain operations and little \(n\)-cubes, J. Am. Math. Soc., 16, 681-704 (2003) · Zbl 1014.18005
[11] Meurer, A. et al.: SymPy: symbolic computing in Python, PeerJ Computer Science e103 (2017); doi:10.7717/peerj-cs.103; software available at http://www.sympy.org
[12] Munkholm, HJ, The Eilenberg-Moore spectral sequence and strongly homotopy multiplicative maps, J. Pure Appl. Algebra, 5, 1-50 (1974) · Zbl 0294.55011
[13] Sinha, DP; Loday, J-L; Vallette, B., The (non-equivariant) homology of the little disks operad,, Operads 2009 (Luminy, 2009), 253-279 (2013), Paris: Société Mathématique de France, Paris
[14] Stasheff, J., Halperin, S.: Differential algebra in its own rite, pp. 567-577 in: Proceedings of the Advanced Study Institute on Algebraic Topology (Aarhus, 1970), vol. 3, Various Publ. Ser. 13, Mat. Inst., Aarhus Univ., Aarhus 1970 · Zbl 0224.55027
[15] Voronov, AA; Gerstenhaber, M., Higher operations on the Hochschild complex, Funct. Anal. Appl., 29, 1-5 (1995) · Zbl 0849.16010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.