Quantum and Braided Lie Algebras

Abstract

We introduce the notion of a braided Lie algebra consisting of a finite-dimensional vector space \CL equipped with a bracket $[\ ,\ ]:\CL\tens\CL\to \CL$and a Yang-Baxter operator$\Psi:\CL\tens\CL\to \CL\tens\CL$obeying some axioms. We show that such an object has an enveloping braided-bialgebra$U(\CL)$. 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$U(\CL)=B(R)$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$\CL$by braided vector fields, the braided-Killing form and the quadratic Casimir associated to$\CL$. 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$\CL\subset U_q(g)$ given by the quantum adjoint action.Comment: 56 page