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Classification of Regular ParametrizedOne-relation Operads

Published online by Cambridge University Press:  20 November 2018

Murray Bremner
Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, S7M 5E6, Saskatoon, Saskatchewan e-mail:
Vladimir Dotsenko
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:


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.

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Copyright © Canadian Mathematical Society 2017

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