The Lie bialgebra structure of the vector space of cyclic words

TL;DR

This paper provides a combinatorial proof of the Lie bialgebra structure on the vector space of reduced cyclic words, originally introduced by M. Chas, linking algebraic and geometric perspectives.

Contribution

It offers a new combinatorial proof of the Lie bialgebra structure, enhancing understanding of algebraic structures related to curves on surfaces.

Findings

Established the combinatorial proof of the Lie bialgebra structure

Connected algebraic structures with geometric representations

Reinforced the isomorphism between cyclic words and surface curves

Abstract

In this paper we give a combinatory proof of the Lie bialgebra structure presented in the vector space of reduced cyclic words. This structure was introduce by M. Chas in Combinatorial lie bialgebras of curves on surfaces, where the proof of the existence of this Lie bialgebra structure is based on the existence of an isomorphism between the space of reduced cyclic word and the space of curves on a surface.