Abstract view
Classification of Regular Parametrized Onerelation Operads


Published:20170801
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, CINVESTAVIPN, Mexico City, Mexico
Abstract
JeanLouis Loday introduced a class of symmetric operads generated
by one bilinear operation subject to one
relation making each leftnormed product of three elements equal
to a linear combination
of rightnormed 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 onerelation 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 onerelation
operads.
In fact, we prove that each such operad is isomorphic to one
of the following five operads: the leftnilpotent 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 onerelation algebra, algebraic operad, Koszul duality, representation theory of the symmetric group, determinantal ideal, Gröbner basis
parametrized onerelation algebra, algebraic operad, Koszul duality, representation theory of the symmetric group, determinantal ideal, Gröbner basis

MSC Classifications: 
18D50, 13B25, 13P10, 13P15, 15A54, 1704, 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, 1708, 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 1704  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, 1708, 33F10]
