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Classification of Regular Parametrized One-relation Operads

  Published:2017-08-01
 Printed: Oct 2017
  • Murray Bremner,
    Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan
  • Vladimir Dotsenko,
    School of Mathematics, Trinity College Dublin, Dublin, Ireland ; Departamento de Matemáticas, CINVESTAV-IPN, Mexico City, Mexico
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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 relation $((a_1a_2)a_3)=0$, 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 Gröbner bases for determinantal ideals and their radicals.
Keywords: parametrized one-relation algebra, algebraic operad, Koszul duality, representation theory of the symmetric group, determinantal ideal, Gröbner basis parametrized one-relation algebra, algebraic operad, Koszul duality, representation theory of the symmetric group, determinantal ideal, Gröbner basis
MSC Classifications: 18D50, 13B25, 13P10, 13P15, 15A54, 17-04, 17A30, 17A50, 20C30, 68W30 show english descriptions Operads [See also 55P48]
Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10]
Grobner bases; other bases for ideals and modules (e.g., Janet and border bases)
Solving polynomial systems; resultants
Matrices over function rings in one or more variables
Explicit machine computation and programs (not the theory of computation or programming)
Algebras satisfying other identities
Free algebras
Representations of finite symmetric groups
Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10]
18D50 - Operads [See also 55P48]
13B25 - Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10]
13P10 - Grobner bases; other bases for ideals and modules (e.g., Janet and border bases)
13P15 - Solving polynomial systems; resultants
15A54 - Matrices over function rings in one or more variables
17-04 - Explicit machine computation and programs (not the theory of computation or programming)
17A30 - Algebras satisfying other identities
17A50 - Free algebras
20C30 - Representations of finite symmetric groups
68W30 - Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10]
 

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