Classification of regular parametrized one-relation operads

TL;DR

This paper classifies all regular parametrized one-relation operads over an algebraically closed field of characteristic zero, showing they are isomorphic to five well-known operads.

Contribution

It provides a complete classification of regular parametrized one-relation operads, identifying them with five classical operads.

Findings

Five isomorphism classes of regular parametrized one-relation operads identified.

Operads correspond to well-known structures: left-nilpotent, associative, Leibniz, dual Leibniz, and Poisson.

Uses advanced computational algebra techniques for classification.

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: \[ (a_1a_2)a_3=\sum_{\sigma\in S_3}x_\sigma\, a_{\sigma(1)}(a_{\sigma(2)}a_{\sigma(3)})\ ; \] 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 identity $((a_1a_2)a_3)=0$, 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 Gr\"obner bases for determinantal ideals and their radicals.