BRAIDED MATRIX STRUCTURE OF THE SKLYANIN ALGEBRA AND OF THE QUANTUM LORENTZ GROUP

Braided groups and braided matrices are novel algebraic structures liv ing in braided or quasitensor categories. As such they are a generaliz ation of super-groups and super-matrices to the case of braid statisti cs. Here we construct braided group versions of the standard quantum g roups U(q)(g). They have the same FRT generators l+/- but a matrix bra ided-coproduct DELTAL = LxL, where L = l+Sl-, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphi c to the braided matrices BM(q)(2); it is a braided-commutative bialge bra in a braided category. As a second application, we show that the q uantum double D(U(q)(sl2)) (also known as the ''quantum Lorentz group' ') is the semidirect product as an algebra of two copies of U(q)(sl2), and also a semidirect product as a coalgebra if we use braid statisti cs. We find various results of this type for the doubles of general qu antum groups and their semi-classical limits as doubles of the Lie alg ebras of Poisson Lie groups.