Compatible associative products and trees

Abstract

We compute dimensions and characters of the components of the operad of two compatible associative products and give an explicit combinatorial construction of the corresponding free algebras in terms of planar rooted trees.

Compatible associative bialgebras

“…In particular, the associative algebras (As 2 (V ), ·) and (As 2 (V ), •) are free as associative algebras. This result is an alternative proof to that obtained by Dotsenko in [3], using operad theory . [12], Theorem 2.6 , we obtain that H is isomorphism to T (PrimH).…”

“…We give an explicit construction of free objects in the category of As 2algebras, easier to work with than the one described in [3]. Using it, we describe a canonical coproduct ∆ on any free As 2 -algebra, which satisfies the unital infinitesimal condition with both products.…”

Free Brace Algebras are Free Pre-Lie Algebras

“…This implies that PR ab is a cofree coalgebra and we recover in a different way the result of freeness of r as a Lie algebra in characteristic zero. Note that a similar result for algebras with two compatible associative products is proved with the same pattern in [6].…”

Totally compatible associative and Lie dialgebras, tridendriform algebras and PostLie algebras

“…A compatible dialgebra is also called an algebra with two compatible associative products [8,27]. The term compatible comes from the fact that, for a k-module V with two associative products * and ⊙, (V, * , ⊙) is a compatible dialgebra if and only if linear combinations of * and ⊙ are still associative.…”

“…Compatible associative products are studied in [22,23,25] in connection with Cartan matrices of affine Dynkin diagrams, integrable matrix equations, infinitesimal bialgebras and quiver representations. In this case, the corresponding operad and free objects are obtained in [8]. More general compatible products are defined in [27].…”