On the structure of the centralizer of a braid

The mixed braid groups are the subgroups of Artin braid groups whose elements preserve a given partition of the base points. We prove that the centralizer of any braid can be expressed in terms of semidirect and direct products of mixed braid groups. Then we construct a generating set of the centralizer of any braid on n strands, which has at most k(k+1)/2 elements if n=2k, and at most $k(k+3)/2 elements if n=2k+1. These bounds are shown to be sharp, due to work of N.V.Ivanov and of S.J.Lee. Finally, we describe how one can explicitly compute this generating set.