Gröbner bases for operads

Abstract

We define a new monoidal structure on the category of collections (shuffle composition). Monoids in this category (shuffle operads) turn out to bring a new insight in the theory of symmetric operads. For this category, we develop the machinery of Gröbner bases for operads and present operadic versions of Bergman's diamond lemma and Buchberger's algorithm. This machinery can be applied to study symmetric operads. In particular, we obtain an effective algorithmic version of Hoffbeck's Poincaré-BirkhoffWitt criterion of Koszulness for (symmetric) quadratic operads.

Operadic Gröbner Bases: An Implementation

“…For exact definitions, we refer the reader to [1,3]. For the purpose of this short communication, we shall concentrate on a less precise definition of symmetric, and shuffle operads.…”

“…For symmetric operads, all permutations are allowed, while shuffle operads have a smaller class of admissible permutations: the shuffle permutations, as described in [1]. The rearranging can either be imagined by allowing branches to cross as the tree is drawn at; or by requiring a planar drawing of each tree, but instead decorating all leaves with integers, and allowing the permutations to act on these integers.…”

“…This universal algebra is captured by the operad defined as a quotient of the free shuffle operad on a single generator by the ideal generated by the two tree polynomials on the left. These give rise to an S-polynomial (using the PathPerm ordering described in [3]) -the one consisting of only the monomial on the right; which will not reduce further. m2 ( 2 , 3 ) ) , +1 % 1 * m2(m2 ( 1 , 2 ) , 3 ) +1 % 1 * m2 ( 1 ,m2 ( 2 , 3 ) ) , +2 % 1 * m2 ( 1 ,m2( 2 ,m2( 3 , 4 ) ) ) , ] m(a, c), b) = m(a, m(b, c)).…”

“…In an upcoming paper [1], the first author and Anton Khoroshkin define the concept of a Gröbner basis for finitely presented operads, prove the diamond lemma for these Gröbner bases, and demonstrate that having a quadratic Gröbner basis is equivalent to the existence of a Poincaré-Birkhoff-Witt basis. As demonstrated by Eric Hoffbeck [2], an operad with a PBW basis is Koszul.…”

“…The authors have, in [3], provided a computer implementation of the algorithms specified by Dotsenko and Khoroshkin. At the core of the paper [1] lies the recognition that every symmetric operad can be thought of as a shuffle operad, and forgetting "unnecessary" symmetries does not affect relevant results of linear and homological algebra for operads; therefore, the respective computation may be restricted to the simpler category of shuffle operads without restricting any conclusions drawn.…”