Classification of Regular ParametrizedOne-relation Operads

Bremner, Murray ; Dotsenko, Vladimir

Canadian journal of mathematics, Tome 69 (2017) no. 5, pp. 992-1035

Résumé

Jean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: $({{a}_{1}}{{a}_{2}}){{a}_{3}}\,=\,\sum{{{_{\sigma }}_{\in {{s}_{3}}}}\,}{{x}_{\sigma }}\,{{a}_{\sigma \,(1)}}({{a}_{\sigma (2)\,}}{{a}_{\sigma (3)}})$ . Such an operad is called a parametrized one-relation operad. For a particular choice of parameters $\{{{x}_{\sigma }}\}$ , this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space $V$ is, as a graded vector space, isomorphic to the tensor algebra of $V$ . We classify, over an algebraically closed field of characteristic zero, all regular parametrized one-relation operads. In fact, we prove that each such operad is isomorphic to one of the following five operads: the left-nilpotent operad defined by the relation $(({{a}_{1}}{{a}_{2}}){{a}_{3}})\,=\,0$ , the associative operad, the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our computational methods combine linear algebra over polynomial rings, representation theory of the symmetric group, and Gröbner bases for determinantal ideals and their radicals.

Détail

DOI : 10.4153/CJM-2017-018-3

Mots-clés : 18D50, 13B25, 13P10, 13P15, 15A54, 17-04, 17A30, 17A50, 20C30, 68W30, parametrized one-relation algebra, algebraic operad, Koszul duality, representation theory of the symmetric group, determinantal ideal, Gröbner basis

Bibliographie

[1] [1] Albert, A. A., Power-associative rings. Trans. Amer. Math. Soc. 64(1948), 552–593. Google Scholar | DOI

[2] [2] Bergman, G. M., The diamond lemma for ring theory. Adv. in Math. 29(1978), no. 2,178–218. http://dx.doi.Org/10.1016/0001-8708(78)90010-5 Google Scholar

[3] [3] Boocher, A., Free resolutions and sparse determinantal ideals. Math. Res. Lett. 19(2012), no. 4, 805–821. http://dx.doi.Org/10.4310/MRL.2012.v1 9.n4.a6 Google Scholar

[4] [4] Bosma, W., Cannon, J., and Playoust, C., The Magma algebra system. I. The user language. J. Symbolic Comput, 24(1997), 235–265. http://dx.doi.Org/10.1006/jsco.1996.0125 Google Scholar

[5] [5] Bremner, M. R. and Dotsenko, V., Algebraic operads: an algorithmic companion. To appear: Chapman and Hall / CRC Press, 2016. Google Scholar

[6] [6] Bremner, M. R., Online addendum to the article “Classification of regular parametrized one-relation operads”. Available via the http://www.maths.tcd.ie/~vdots/BDaddendum.pdf, along with the copy-pasteable Magma script http://www.maths.tcd.ie/~vdots/BDaddendum.txt. Google Scholar

[7] [7] Bremner, M. R., Madariaga, S., and Peresi, L. A., Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions. Comment. Math. Univ. Carolin. 57(2016), no. 4, 413–452. Google Scholar

[8] [8] Clifton, J. M., A simplification of the computation of the natural representation of the symmetric group S. Proc. Amer. Math. Soc. 83(1981), no. 2, 248–250. Google Scholar

[9] [9] Cox, D., Little, J., and O'Shea, D., Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. 3rd ed. Undergraduate Texts in Mathematics, 3. Springer, New York, 2007. Google Scholar | DOI

[10] [10] Cox, D., Using algebraic geometry. 2nd ed. Graduate Texts in Mathematics, 185. Springer, New York, 2005. Google Scholar

[11] [11] Dotsenko, V. and Khoroshkin, A., Gröbner bases for operads. Duke Math. J. 153(2010), no. 2, 363–396. Google Scholar | DOI

[12] [12] Drinfel'd, V. G., On quadratic commutation relations in the quasiclassical case. Selecta Math. Soviet. 11(1992), no. 4, 317–326. Google Scholar

[13] [13] Ginzburg, V., and Kapranov, M., Koszul duality for operads. Duke Math. J. 76(1994), no. 1, 203–272. Google Scholar | DOI

[14] [14] Livernet, M., Dualité de Koszul pour les opérades quadratiques binaires. M.Sc. thesis, Université de Strasbourg, 1995. Google Scholar

[15] [15] Loday, J.-L. and Vallette, B., Algebraic operads. Grundlehren der Mathematischen Wissenschaften, 346. Springer, Heidelberg, 2012. Google Scholar

[16] [16] Magma Calculator, http://magma.maths.usyd.edu.au/calc/. Google Scholar

[17] [17] Markl, M. and Remm, E., Algebras with one operation including Poisson and other Lie-admissible algebras. J. Algebra 299(2006), no. 1,171–189. http://dx.doi.Org/10.1016/j.jalgebra.2OO5.O9.O18 Google Scholar

[18] [18] Markl, M., (Non-)Koszulness of operads for n-ary algebras, galgalim and other curiosities. J. Homotopy Relat. Struct. 10(2015) no. 4, 939–969. http://dx.doi.Org/10.1007/s40062-014-0090-7 Google Scholar

[19] [19] Miro-Roig, R. M., Determinantal deals. Progress in Mathematics, 264. Birkhäuser Verlag, Basel, 2008. Google Scholar

[20] [20] Polishchuk, A. and Positselski, L., Quadratic algebras. American Mathematical Society, Providence, RI, 2005. http://dx.doi.Org/10.1090/ulect/037 Google Scholar

[21] [21] Stanley, R. P., Catalan numbers. Cambridge University Press, New York, 2015. http://dx.doi.Org/10.1017/CBO9781139871 495 Google Scholar

[22] [22] Zinbiel, G. W., Encyclopedia of types of algebras 2010. In: Operads and universal algebra. World Scientific Publishing, Hackensack, NJ, 2012, pp. 217–297. Google Scholar | DOI