Braided Lie algebras and bicovariant differential calculi over coquasitriangular Hopf algebras

TL;DR

This paper explores the structure of braided Lie algebras derived from bicovariant differential calculi over coquasitriangular Hopf algebras, revealing new algebraic properties and functors with applications to quantum groups.

Contribution

It introduces a braided Lie algebra framework for quantum tangent spaces and constructs a quantum Lie functor with classical-like properties for coquasitriangular Hopf algebras.

Findings

The quantum universal enveloping algebra is quadratic for inner calculi.

The quantum Lie functor yields trivial results on standard quantum groups.

Constructed generalized-Lie algebras for $O_q(SL_n)$, recovering the Witten algebra for $n=2$.

Abstract

We show that if $g_\Gamma$ is the quantum tangent space (or quantum Lie algebra in the sense of Woronowicz) of a bicovariant first order differential calculus over a coquasitriangular Hopf algebra $(A,r)$, then a certain extension of it is a braided Lie algebra in the category of $A$-comodules. This is used to show that the Woronowicz quantum universal enveloping algebra $U(g_\Gamma)$ is a bialgebra in the braided category of $A$-comodules. We show that this algebra is quadratic when the calculus is inner. Examples with this unexpected property include finite groups and quantum groups with their standard differential calculi. We also find a quantum Lie functor for coquasitriangular Hopf algebras, which has properties analogous to the classical one. This functor gives trivial results on standard quantum groups $O_q(G)$, but reasonable ones on examples closer to the classical case, such as the cotriangular Jordanian deformations. In addition, we show that split braided Lie algebras define `generalised-Lie algebras' in a different sense of deforming the adjoint representation. We construct these and their enveloping algebras for $O_q(SL_n)$, recovering the Witten algebra for $n=2$.