Quantum and Braided Lie Algebras

TL;DR

This paper introduces the concept of braided Lie algebras, generalizing classical Lie algebras with a braiding structure, and explores their properties, enveloping algebras, and connections to quantum groups.

Contribution

It defines braided Lie algebras with axioms, constructs their enveloping braided-bialgebras, and links them to quantum groups and classical Lie algebra structures.

Findings

Braided Lie algebras generalize classical, color, and super-Lie algebras.

Enveloping algebra $U( ext{CL})$ exists for braided Lie algebras.

Standard quantum groups $U_q(g)$ are examples of these structures.

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.