QUANTUM AND BRAIDED-LIE ALGEBRAS

We introduce the notion of a braided-Lie algebra consisting of a finit e-dimensional vector space L equipped with a bracket [ , ]: L X L --> L and a Yang-Baxter operator PSI: L X L --> L obeying some axioms. We show that such an object has an enveloping braided-bialgebra U(L). We show that every generic R-matrix leads to such a braided-Lie algebra w ith [ , ] given by structure constants c(IJ)K determined from R. In th is case U(L) = B(R) the braided matrices introduced previously. We als o introduce the basic theory of these braided-Lie algebras, including the natural right-regular action of a braided-Lie algebra L by braided vector fields, the braided-Killing form and the quadratic Casimir ass ociated to L. These constructions recover the relevant notions for usu al, colour and super-Lie algebras as special cases. In addition, the s tandard quantum deformations U(q)(g) are understood as the enveloping algebras of such underlying braided-Lie algebras with [ , ] on L subse t-of U(q) (g) given by the quantum adjoint action.