82. 12/08/2024 Pierre Catoire (University of the Littoral Opal Coast, France)

Title: The free tridendriform algebra, Schroeder trees and Hopf algebras

Abstract: The notions of dendriform algebras, respectively tridendriform, describe the action of some elements of the symmetric groups called shuffle, respectively quasi-shuffle over the set of words whose letters are elements of an alphabet, respectively of a monoid. A link between dendriform and tridendriform algebras will be made. Those words algebras satisfy some properties but they are not free. This means that they satisfy extra properties like commutativity. In this talk, we will describe the free tridendriform algebra. It will be described with planar trees (not necessarily binary) called Schroeder trees. We will describe the tridendriform structure over those trees in a non-recursive way. Then, we will build a coproduct on this algebra that will make it a (3, 2)-dendriform bialgebra graded by the number of leaves. Once it will be build, we will study this Hopf algebra: duality, quotient spaces, dimensions, study of the primitive elements.

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