The associative filtration of the dendriform operad

Abstract

The associative filtration of the dendriform operad Norah Mohammed Alghamdi A dendriform algebra is a vector space V with two binary operations denoted < and > that satisfy the following three algebraic properties for all elements a1, a2, a3 of V : (a1 > a2) < a3 = a1 > (a2 < a3), (a1 < a2) < a3 = a1 < (a2 < a3 + a2 > a3), (a1 < a2 + a1 > a2) > a3 = a1 > (a2 > a3). It is well known that the sum of the two operations in any dendriform algebra, the operation a1 ?a2 = a1 < a2+a1 > a2, is always associative. We consider the filtration of the nonsymmetric operad Dend by powers of the ideal generated by this associative operation, and the associated graded operad. For pre-Lie algebras, a similar question was considered in a recent paper of Dotsenko, where the associated graded operad was related to the so called F-manifolds. Similarly to the pre-Lie case, the associated graded operad of the dendriform operad turns out to be presented by quadratic and cubic relations. However, the cubic relations have more complicated structure than the ones found by Dotsenko in the pre-Lie case, so his approach is not applicable. However, we were able to make more use of operadic Gröbner bases than it is possible in the pre-Lie case, leading to a complete description of the associated graded operad.