Manin triples and differential operators on quantum groups

TL;DR

This paper explores the connection between differential operators on quantum groups at roots of unity and Poisson structures, revealing their equivalence with structures introduced via Manin triples.

Contribution

It demonstrates that the quasi-classical limit of differential operators on quantized algebraic groups aligns with Poisson structures from Manin triples, bridging quantum and classical frameworks.

Findings

Poisson manifold obtained from differential operators at roots of 1

Equivalence of this Poisson structure with Semenov-Tyan-Shansky's structure

Connection established between quantum group operators and classical Poisson geometry

Abstract

By taking the quasi-classical limit of the ring of differential operators on a quantized algebraic group at roots of 1 we obtain a certain Poisson manifold. We show that this Poisson structure coincides with the one introduced by Semenov-Tyan-Shansky geometrically in the framework of Manin triples.