Deformation of the Hopf algebra of plane posets

TL;DR

This paper introduces a four-parameter deformation of the Hopf algebra of plane posets, resulting in a family of braided, self-dual Hopf algebras, and connects it to deformations of free quasi-symmetric functions.

Contribution

It presents a novel four-parameter deformation of the Hopf algebra of plane posets and establishes an isomorphism with deformations of free quasi-symmetric functions in a special case.

Findings

Family of braided, self-dual Hopf algebras derived from the deformation.

Identification of a special case isomorphic to a deformation of free quasi-symmetric functions.

Extension of the algebraic structure of plane posets through deformation parameters.

Abstract

We describe and study a four parameters deformation of the two products and the coproduct of the Hopf algebra of plane posets. We obtain a family of braided Hopf algebras, generally self-dual. We also prove that in a particular case (when the second parameter goes to zero and the first and third parameters are equal), this deformation is isomorphic, as a self-dual braided Hopf algebra, to a deformation of the Hopf algebra of free quasi-symmetric functions.