SOLUTIONS OF THE YANG-BAXTER EQUATIONS FROM BRAIDED-LIE ALGEBRAS AND BRAIDED GROUPS

We obtain an R-matrix or matrix representation of the Artin braid grou p acting in a canonical way on the vector space of every (super)-Lie a lgebra or braided-Lie algebra. The same result applies for every (supe r)-Hopf algebra or braided-Hopf algebra. We recover some known represe ntations such as those associated to racks. We also obtain new represe ntations such as a non-trivial one on the ring k[x] of polynomials in one variable, regarded as a braided-line. Representations of the exten ded Artin braid group for braids in the complement of S-1 are also obt ained by the same method.