On some groups related to the Braid Groups of type A

Abstract

We prove that a family of groups R(n) forms the algebraic structure of an operad and that they admit a presentation similar to that of the Braid groups of type A.  This result provides a new proof that the Braid Groups form an operad, a topic emphasized in ~\cite{16}~\cite{ulrike}. These groups proved to be useful in several problems which belong to different areas of Mathematics. Representations of R(n) came from a system of mixed Yang-Baxter type equations. We define the Hopf equation in braided monoidal categories and we prove that representations for our groups came from any braided Hopf algebra with invertible antipode. Using this result, we prove that there is a morphism from R(n) to the mapping class group $\Gamma_{n,1}$, using some results from 3-dimensional topology.