Braided Lie algebras and bicovariant differential calculi over coquasitriangular Hopf algebras
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 `generalisedLie 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 $n=2$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2001
 arXiv:
 arXiv:math/0112299
 Bibcode:
 2001math.....12299G
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Algebraic Geometry;
 High Energy Physics  Theory
 EPrint:
 42 pages latex