Skip to main content Accessibility help



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.



Hide All
1.Ataguema, H. and Makhlouf, A., Notes on cohomologies of ternary algebras of associative type. J. Gen. Lie Theory Appl. 3(3) (2009), 157174.
2.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).
3.Bremner, M., On free partially associative triple systems. Comm. Algebra 28(4) (2000), 21312145.
4.Bremner, M. and Dotsenko, V., Algebraic operads: an algorithmic companion (CRC Press, Boca Raton, FL, 2016).
5.Carlsson, R., Cohomology of associative triple systems. Proc. Amer. Math. Soc. 60 (1976), 17. Erratum and supplement: Proc. Amer. Math. Soc. 67(2) (1977), 361.
6.Carlsson, R., n-ary algebras. Nagoya Math. J . 78 (1980), 4556.
7.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).
8.Drummond-Cole, G. and Vallette, B., The minimal model for the Batalin-Vilkovisky operad. Selecta Math. (N.S.) 19(1) (2013), 147.
9.Eilenberg, S. and Mac Lane, S., Cohomology theory in abstract groups, I. Annals of Math . 48 (1947), 5178.
10.Gerstenhaber, M., The cohomology structure of an associative ring. Ann. Math. 78(2) (1963), 267288.
11.Gerstenhaber, M., On the deformation of rings and algebras. Ann. Math. 79(2) (1964), 59103.
12.Gerstenhaber, M. and Voronov, A., Higher-order operations on the Hochschild complex. Funktsional. Anal. i Prilozhen 29(1) (1995), 16, 96. Translation: Funct. Anal. Appl. 29(1) (1995), 1–5.
13.Getzler, E., Batalin-Vilkovisky algebras and two-dimensional topological field theories. Comm. Math. Phys. 159(2) (1994), 265285.
14.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).
15.Goze, N. and Remm, E., Dimension theorem for free ternary partially associative algebras and applications. J. Algebra 348 (2011), 1436.
16.Hestenes, M., A ternary algebra with applications to matrices and linear transformations. Arch. Rational Mech. Anal. 11 (1962), 138194.
17.Hochschild, G., On the cohomology groups of an associative algebra. Ann. Math. 46(2) (1945), 5867.
18.Huebschmann, J., Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras. Ann. Inst. Fourier (Grenoble) 48(2) (1998), 425440.
19.Lister, W., Ternary rings. Trans. Amer. Math. Soc. 154 (1971), 3755.
20.Loday, J.-L., Cup-product for Leibniz cohomology and dual Leibniz algebras. Math. Scand. 77(2) (1995), 189196.
21.Loday, J.-L. and Vallette, B., Algebraic operads. Grundlehren Math. Wiss, vol. 346 (Springer, Heidelberg, 2012).
22.Loos, O., Tripelsysteme, Assoziative. Manuscripta Math . 7 (1972), 103112.
23.Markl, M., Models for operads. Comm. Algebra 24(4) (1996), 14711500.
24.Markl, M., Cohomology operations and the Deligne conjecture. Czechoslovak Math. J . 57(1) (2007), 473503.
25.Markl, M. and Remm, E.: Operads for n-ary algebras: calculations and conjectures. Arch. Math. (Brno) 47(5) (2011), 377387.
26.Markl, M. and Remm, E., (Non-)Koszulness of operads for n-ary algebras, galgalim and other curiosities. J. Homotopy Relat. Struct . 10(4) (2015), 939969.
27.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).
28.McClure, J. and Smith, J., Multivariable cochain operations and little n-cubes. J. Am. Math. Soc. 16 (2003), 681704.
29.Shukla, U., Cohomologie des algèbres associatives. Ann. Sci. École Norm. Sup. 78(3) (1961), 163209.
30.Weibel, C., History of homological algebra, History of Topology (North-Holland, Amsterdam, 1999), 797836.

MSC classification



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.