Bicovariant Quantum Algebras and Quantum Lie Algebras

TL;DR

This paper develops a bicovariant calculus of differential operators on quantum groups, revealing their algebraic structure as generalized Lie algebras and providing computational tools and applications like quantum determinants.

Contribution

It introduces a natural construction of bicovariant differential operators on quantum groups using braid group elements, connecting Hopf algebra and matrix formulations.

Findings

Operators form generalized Lie algebras under adjoint action

Quantum determinant for $SO_q(N)$ derived

Orthogonality relation for the reflection matrix $Y$ established

Abstract

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given by elements of the pure braid group. These operators --- the `reflection matrix' $Y \equiv L^+ SL^-$ being a special case --- generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation for $Y$ in $SO_q(N)$.