The quantum duality principle

TL;DR

This paper provides a detailed proof of the quantum duality principle, establishing a correspondence between quantizations of Lie bialgebras via quantum universal enveloping algebras and quantum formal series Hopf algebras.

Contribution

It offers the first complete proof of the quantum duality principle, confirming the duality between QUEAs and QFSHAs as claimed by Drinfeld.

Findings

Established functors between QUEA and QFSHA are inverses

Confirmed the duality of Lie bialgebras via quantization methods

Provided a rigorous proof of a previously unproven claim

Abstract

The "quantum duality principle" states that the quantization of a Lie bialgebra - via a quantum universal enveloping algebra (QUEA) - provides also a quantization of the dual Lie bialgebra (through its associated formal Poisson group) - via a quantum formal series Hopf algebra (QFSHA) - and, conversely, a QFSHA associated to a Lie bialgebra (via its associated formal Poisson group) yields a QUEA for the dual Lie bialgebra as well; more precisely, there exist functors QUEA --> QFSHA and QFSHA --> QUEA, inverse of each other, such that in either case the Lie bialgebra associated to the target object is the dual of that of the source object. Such a result was claimed true by Drinfeld, but seems to be unproved in literature: we give here a complete detailed proof of it.