Braided Differential Operators on Quantum Algebras

TL;DR

This paper introduces braided differential algebras as a quantum analog of differential operator algebras on algebraic varieties with group actions, exemplified by gl^*(m) and coadjoint orbits, including the Heisenberg double of Fun_q(GL(m)).

Contribution

It defines a new class of braided differential algebras that generalize classical differential operators in a quantum setting, with explicit examples and connections to quantum groups.

Findings

Construction of braided differential algebras as quantizations.

Application to gl^*(m) and coadjoint orbits.

Relation to the Heisenberg double of quantum groups.

Abstract

We define the braided differential algebras which can be interpreted as quantization of the differential operator algebra defined on some algebraic varieties supplied with the action of the group GL(m). The algebra is generated by right invariant or coajoint vector fields. Our main example is gl^*(m) and coadjoint orbits in it. The Heisenberg double on the quantum group Fun_q(GL(m)is a particular case of the suggested construction.