Quantum and Braided Lie Algebras

We introduce the notion of a braided Lie algebra consisting of a finite-dimensional vector space L equipped with a bracket [ , ] : L⊗L → L and a Yang-Baxter operator Ψ : L⊗L → 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 with [ , ] given by structure constants cK determined from R. In this case U(L) = B(R) the braided matrices introduced previously. We also 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 associated to L. These constructions recover the relevant notions for usual, colour and super-Lie algebras as special cases. In addition, the standard quantum deformations Uq(g) are understood as the enveloping algebras of such underlying braided Lie algebras with [ , ] on L ⊂ Uq(g) given by the quantum adjoint action.