Skip to main content Accessibility help
×
Home
Hostname: page-component-684bc48f8b-b5g75 Total loading time: 0.784 Render date: 2021-04-11T04:27:14.070Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Classification of Regular ParametrizedOne-relation Operads

Published online by Cambridge University Press:  20 November 2018

Murray Bremner
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, S7M 5E6, Saskatoon, Saskatchewan e-mail: bremner@math.usask.ca
Vladimir Dotsenko
Affiliation:
School of Mathematics, Trinity College Dublin, College Green, Dublin 2, Ireland and Departamento de Matemáticas, CINVESTAV-IPN, Av., Instituto Politécnico Nacional 2508, Col. San Pedro Zacatenco, México, D.F., CP 07360, Mexico e-mail: vdots@maths.tcd.ie
Corresponding

Abstract

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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2017

Access options

Get access to the full version of this content by using one of the access options below.

References

[1] Albert, A. A., Power-associative rings. Trans. Amer. Math. Soc. 64(1948), 552593.http://dx.doi.org/10.1090/S0002-9947-1948-0027750-7 CrossRefGoogle Scholar
[2] Bergman, G. M., The diamond lemma for ring theory. Adv. in Math. 29(1978), no. 2,178218.http://dx.doi.Org/10.1016/0001-8708(78)90010-5 CrossRefGoogle Scholar
[3] Boocher, A., Free resolutions and sparse determinantal ideals. Math. Res. Lett. 19(2012), no. 4, 805821.http://dx.doi.Org/10.4310/MRL.2012.v1 9.n4.a6 CrossRefGoogle Scholar
[4] Bosma, W., Cannon, J., and Playoust, C., The Magma algebra system. I. The user language. J. Symbolic Comput, 24(1997), 235265.http://dx.doi.Org/10.1006/jsco.1996.0125 CrossRefGoogle Scholar
[5] Bremner, M. R. and Dotsenko, V., Algebraic operads: an algorithmic companion. To appear: Chapman and Hall / CRC Press, 2016.Google Scholar
[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] 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, 413452.Google Scholar
[8] Clifton, J. M., A simplification of the computation of the natural representation of the symmetric group Sn. Proc. Amer. Math. Soc. 83(1981), no. 2, 248250.Google Scholar
[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. http://dx.doi.org/10.1007/978-0-387-35651-8 Google Scholar
[10] Cox, D., Using algebraic geometry. 2nd ed. Graduate Texts in Mathematics, 185. Springer, New York, 2005.Google Scholar
[11] Dotsenko, V. and Khoroshkin, A., Gröbner bases for operads. Duke Math. J. 153(2010), no. 2, 363396.http://dx.doi.org/10.1215/00127094-2010-026 CrossRefGoogle Scholar
[12] Drinfel'd, V. G., On quadratic commutation relations in the quasiclassical case. Selecta Math. Soviet. 11(1992), no. 4, 317326.Google Scholar
[13] Ginzburg, V., and Kapranov, M., Koszul duality for operads. Duke Math. J. 76(1994), no. 1, 203272.http://dx.doi.org/10.1215/S0012-7094-94-07608-4 CrossRefGoogle Scholar
[14] Livernet, M., Dualité de Koszul pour les opérades quadratiques binaires. M.Sc. thesis, Université de Strasbourg, 1995.Google Scholar
[15] Loday, J.-L. and Vallette, B., Algebraic operads. Grundlehren der Mathematischen Wissenschaften, 346. Springer, Heidelberg, 2012.Google Scholar
[17] Markl, M. and Remm, E., Algebras with one operation including Poisson and other Lie-admissible algebras. J. Algebra 299(2006), no. 1,171189.http://dx.doi.Org/10.1016/j.jalgebra.2OO5.O9.O18 CrossRefGoogle Scholar
[18] Markl, M., (Non-)Koszulness of operads for n-ary algebras, galgalim and other curiosities. J. Homotopy Relat. Struct. 10(2015) no. 4, 939969.http://dx.doi.Org/10.1007/s40062-014-0090-7 CrossRefGoogle Scholar
[19] Miro-Roig, R. M., Determinantal deals. Progress in Mathematics, 264. Birkhäuser Verlag, Basel, 2008.Google Scholar
[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] Stanley, R. P., Catalan numbers. Cambridge University Press, New York, 2015.http://dx.doi.Org/10.1017/CBO9781139871 495 Google Scholar
[22] Zinbiel, G. W., Encyclopedia of types of algebras 2010. In: Operads and universal algebra. World Scientific Publishing, Hackensack, NJ, 2012, pp. 217297. http://dx.doi.org/10.1142/978981 4365123J3O11 Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 22 *
View data table for this chart

* Views captured on Cambridge Core between 20th November 2018 - 11th April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Classification of Regular Parametrized One-relation Operads
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Classification of Regular Parametrized One-relation Operads
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Classification of Regular Parametrized One-relation Operads
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *