The Lie bialgebra structure of the vector space of cyclic words

This paper presents a combinatorial proof of the existence of a Lie bialgebra structure over the vector space of reduced cyclic words. Any surface with non-empty boundary has an associated vector space determined by the corresponding surface symbol, this space is known as the space of reduced cyclic words. The Lie bialgebra structure over this space was introduced by Chas in the article Combinatorial Lie bialgebras of curves on surfaces, where a proof of the existence of this structure is given. This proof is based on the construction of an isomorphism between the space of reduced cyclic words and the space of curves on a surface.