Abstract
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 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 c IJ K 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 U q(g) are understood as the enveloping algebras of such underlying braided-Lie algebras with [ , ] on L ⊂ U q(g) given by the quantum adjoint action.
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