Geodesic
OPERADIC APPROACH TO COHOMOLOGY OF ASSOCIATIVE TRIPLE AND N-TUPLE SYSTEMS
Glasgow mathematical journal, Tome 62 (2020), pp. S128-S141

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
BAGHERZADEH, FATEMEH; BREMNER, MURRAY. OPERADIC APPROACH TO COHOMOLOGY OF ASSOCIATIVE TRIPLE AND N-TUPLE SYSTEMS. Glasgow mathematical journal, Tome 62 (2020), pp. S128-S141. doi: 10.1017/S0017089519000454
@article{10_1017_S0017089519000454,
     author = {BAGHERZADEH, FATEMEH and BREMNER, MURRAY},
     title = {OPERADIC {APPROACH} {TO} {COHOMOLOGY} {OF} {ASSOCIATIVE} {TRIPLE} {AND} {N-TUPLE} {SYSTEMS}},
     journal = {Glasgow mathematical journal},
     pages = {S128--S141},
     year = {2020},
     volume = {62},
     number = {S1},
     doi = {10.1017/S0017089519000454},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089519000454/}
}
TY  - JOUR
AU  - BAGHERZADEH, FATEMEH
AU  - BREMNER, MURRAY
TI  - OPERADIC APPROACH TO COHOMOLOGY OF ASSOCIATIVE TRIPLE AND N-TUPLE SYSTEMS
JO  - Glasgow mathematical journal
PY  - 2020
SP  - S128
EP  - S141
VL  - 62
IS  - S1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089519000454/
DO  - 10.1017/S0017089519000454
ID  - 10_1017_S0017089519000454
ER  - 
%0 Journal Article
%A BAGHERZADEH, FATEMEH
%A BREMNER, MURRAY
%T OPERADIC APPROACH TO COHOMOLOGY OF ASSOCIATIVE TRIPLE AND N-TUPLE SYSTEMS
%J Glasgow mathematical journal
%D 2020
%P S128-S141
%V 62
%N S1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089519000454/
%R 10.1017/S0017089519000454
%F 10_1017_S0017089519000454

[1] Ataguema, H. and Makhlouf, A., Notes on cohomologies of ternary algebras of associative type. J. Gen. Lie Theory Appl. 3(3) (2009), 157–174. Google Scholar | DOI

[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). Google Scholar

[3] Bremner, M., On free partially associative triple systems. Comm. Algebra 28(4) (2000), 2131–2145. Google Scholar | DOI

[4] Bremner, M. and Dotsenko, V., Algebraic operads: an algorithmic companion (CRC Press, Boca Raton, FL, 2016). Google Scholar | DOI

[5] 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. Google Scholar | DOI

[6] Carlsson, R., n-ary algebras. Nagoya Math. J . 78 (1980), 45–56. Google Scholar | DOI

[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). Google Scholar

[8] Drummond-Cole, G. and Vallette, B., The minimal model for the Batalin-Vilkovisky operad. Selecta Math. (N.S.) 19(1) (2013), 1–47. Google Scholar | DOI

[9] Eilenberg, S. and Mac Lane, S., Cohomology theory in abstract groups, I. Annals of Math . 48 (1947), 51–78. Google Scholar | DOI

[10] Gerstenhaber, M., The cohomology structure of an associative ring. Ann. Math. 78(2) (1963), 267–288. Google Scholar

[11] Gerstenhaber, M., On the deformation of rings and algebras. Ann. Math. 79(2) (1964), 59–103. Google Scholar | DOI

[12] 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

[13] Getzler, E., Batalin-Vilkovisky algebras and two-dimensional topological field theories. Comm. Math. Phys. 159(2) (1994), 265–285. Google Scholar | DOI

[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). Google Scholar | DOI

[15] Goze, N. and Remm, E., Dimension theorem for free ternary partially associative algebras and applications. J. Algebra 348 (2011), 14–36. Google Scholar | DOI

[16] Hestenes, M., A ternary algebra with applications to matrices and linear transformations. Arch. Rational Mech. Anal. 11 (1962), 138–194. Google Scholar | DOI

[17] Hochschild, G., On the cohomology groups of an associative algebra. Ann. Math. 46(2) (1945), 58–67. Google Scholar | DOI

[18] Huebschmann, J., Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras. Ann. Inst. Fourier (Grenoble) 48(2) (1998), 425–440. Google Scholar | DOI

[19] Lister, W., Ternary rings. Trans. Amer. Math. Soc. 154 (1971), 37–55. Google Scholar | DOI

[20] Loday, J.-L., Cup-product for Leibniz cohomology and dual Leibniz algebras. Math. Scand. 77(2) (1995), 189–196. Google Scholar | DOI

[21] Loday, J.-L. and Vallette, B., Algebraic operads. Grundlehren Math. Wiss, vol. 346 (Springer, Heidelberg, 2012). Google Scholar

[22] Loos, O., Tripelsysteme, Assoziative. Manuscripta Math . 7 (1972), 103–112. Google Scholar | DOI

[23] Markl, M., Models for operads. Comm. Algebra 24(4) (1996), 1471–1500. Google Scholar | DOI

[24] Markl, M., Cohomology operations and the Deligne conjecture. Czechoslovak Math. J . 57(1) (2007), 473–503. Google Scholar | DOI

[25] Markl, M. and Remm, E.: Operads for n-ary algebras: calculations and conjectures. Arch. Math. (Brno) 47(5) (2011), 377–387. Google Scholar

[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), 939–969. Google Scholar | DOI

[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). Google Scholar

[28] Mcclure, J. and Smith, J., Multivariable cochain operations and little n-cubes. J. Am. Math. Soc. 16 (2003), 681–704. Google Scholar | DOI

[29] Shukla, U., Cohomologie des algèbres associatives. Ann. Sci. École Norm. Sup. 78(3) (1961), 163–209. Google Scholar | DOI

[30] Weibel, C., History of homological algebra, History of Topology (North-Holland, Amsterdam, 1999), 797–836. Google Scholar | DOI

Cité par Sources :