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OPERADIC APPROACH TO COHOMOLOGY OF ASSOCIATIVE TRIPLE AND N-TUPLE SYSTEMS

Part of: Categories with structure Homological methods General nonassociative rings Rings and algebras arising under various constructions

Published online by Cambridge University Press:  14 November 2019

FATEMEH BAGHERZADEH  and
MURRAY BREMNER Open the ORCID record for MURRAY BREMNER [Opens in a new window]
FATEMEH BAGHERZADEH
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Canada e-mails: [email protected], [email protected]
MURRAY BREMNER
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Canada e-mails: [email protected], [email protected]
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Abstract

The cup product in the cohomology of algebras over quadratic operads has been studied in the general setting of Koszul duality for operads. We study the cup product on the cohomology of n-ary totally associative algebras with an operation of even (homological) degree. This cup product endows the cohomology with the structure of an n-ary partially associative algebra with an operation of even or odd degree depending on the parity of n. In the cases n=3 and n=4, we provide an explicit definition of this cup product and prove its basic properties.

MSC classification

Primary: 18D50: Operads
Secondary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.) 16S37: Quadratic and Koszul algebras 17A42: Other $n$-ary compositions $(n ge 3)$
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Special Issue S1: Workshop on Nonassociative algebras and their applications , December 2020 , pp. S128 - S141
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019

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References

REFERENCES

Ataguema, H. and Makhlouf, A., Notes on cohomologies of ternary algebras of associative type. J. Gen. Lie Theory Appl. 3(3) (2009), 157–174.CrossRefGoogle Scholar
Bagherzadeh, F. and Bremner, M., Gröbner bases and dimension formulas for ternary partially associative operads. To appear in: A. Ambily, R. Hazrat and B. Sury (editors). Leavitt path algebras and classical K-theory, Indian Statistical Institute Series (Springer, 2019).Google Scholar
Bremner, M., On free partially associative triple systems. Comm. Algebra 28(4) (2000), 2131–2145.CrossRefGoogle Scholar
Bremner, M. and Dotsenko, V., Algebraic operads: an algorithmic companion (CRC Press, Boca Raton, FL, 2016).CrossRefGoogle Scholar
Carlsson, R., Cohomology of associative triple systems. Proc. Amer. Math. Soc. 60 (1976), 1–7. Erratum and supplement: Proc. Amer. Math. Soc. 67(2) (1977), 361.CrossRefGoogle Scholar
Carlsson, R., n-ary algebras. Nagoya Math. J . 78 (1980), 45–56.CrossRefGoogle Scholar
Dotsenko, V., Markl, M. and Remm, E., Non-Koszulness of operads and positivity of Poincaré series. arXiv:1604.08580 [math.KT] (submitted on 28 April 2016).Google Scholar
Drummond-Cole, G. and Vallette, B., The minimal model for the Batalin-Vilkovisky operad. Selecta Math. (N.S.) 19(1) (2013), 1–47.CrossRefGoogle Scholar
Eilenberg, S. and Mac Lane, S., Cohomology theory in abstract groups, I. Annals of Math . 48 (1947), 51–78.CrossRefGoogle Scholar
Gerstenhaber, M., The cohomology structure of an associative ring. Ann. Math. 78(2) (1963), 267–288.Google Scholar
Gerstenhaber, M., On the deformation of rings and algebras. Ann. Math. 79(2) (1964), 59–103.CrossRefGoogle Scholar
Gerstenhaber, M. and Voronov, A., Higher-order operations on the Hochschild complex. Funktsional. Anal. i Prilozhen 29(1) (1995), 1–6, 96. Translation: Funct. Anal. Appl. 29(1) (1995), 1–5.Google Scholar
Getzler, E., Batalin-Vilkovisky algebras and two-dimensional topological field theories. Comm. Math. Phys. 159(2) (1994), 265–285.CrossRefGoogle Scholar
Gnedbaye, A., Opérades des algèbres (k+1)-aires, in Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), 83–113. Contemp. Math., vol. 202 (Amer. Math. Soc., Providence, RI, 1997).CrossRefGoogle Scholar
Goze, N. and Remm, E., Dimension theorem for free ternary partially associative algebras and applications. J. Algebra 348 (2011), 14–36.CrossRefGoogle Scholar
Hestenes, M., A ternary algebra with applications to matrices and linear transformations. Arch. Rational Mech. Anal. 11 (1962), 138–194.CrossRefGoogle Scholar
Hochschild, G., On the cohomology groups of an associative algebra. Ann. Math. 46(2) (1945), 58–67.CrossRefGoogle Scholar
Huebschmann, J., Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras. Ann. Inst. Fourier (Grenoble) 48(2) (1998), 425–440.CrossRefGoogle Scholar
Lister, W., Ternary rings. Trans. Amer. Math. Soc. 154 (1971), 37–55.CrossRefGoogle Scholar
Loday, J.-L., Cup-product for Leibniz cohomology and dual Leibniz algebras. Math. Scand. 77(2) (1995), 189–196.CrossRefGoogle Scholar
Loday, J.-L. and Vallette, B., Algebraic operads. Grundlehren Math. Wiss, vol. 346 (Springer, Heidelberg, 2012).Google Scholar
Loos, O., Tripelsysteme, Assoziative. Manuscripta Math . 7 (1972), 103–112.CrossRefGoogle Scholar
Markl, M., Models for operads. Comm. Algebra 24(4) (1996), 1471–1500.CrossRefGoogle Scholar
Markl, M., Cohomology operations and the Deligne conjecture. Czechoslovak Math. J . 57(1) (2007), 473–503.CrossRefGoogle Scholar
Markl, M. and Remm, E.: Operads for n-ary algebras: calculations and conjectures. Arch. Math. (Brno) 47(5) (2011), 377–387.Google Scholar
Markl, M. and Remm, E., (Non-)Koszulness of operads for n-ary algebras, galgalim and other curiosities. J. Homotopy Relat. Struct . 10(4) (2015), 939–969.CrossRefGoogle Scholar
Markl, M., Shnider, S. and Stasheff, J., Operads in Algebra, Topology and Physics, Mathematical Surveys and Monographs, vol. 96 (American Mathematical Society, Providence, RI, 2002).Google Scholar
McClure, J. and Smith, J., Multivariable cochain operations and little n-cubes. J. Am. Math. Soc. 16 (2003), 681–704.CrossRefGoogle Scholar
Shukla, U., Cohomologie des algèbres associatives. Ann. Sci. École Norm. Sup. 78(3) (1961), 163–209.CrossRefGoogle Scholar
Weibel, C., History of homological algebra, History of Topology (North-Holland, Amsterdam, 1999), 797–836.CrossRefGoogle Scholar