The global quantum duality principle: theory, examples, applications

TL;DR

This paper generalizes the quantum duality principle to a broad algebraic setting, establishing a Galois correspondence between quantum function algebras and quantum universal enveloping algebras, with applications to various groups and quantum field theory.

Contribution

It extends the quantum duality principle to torsionless Hopf algebras over a principal ideal domain, providing a systematic method to derive quantum groups from classical ones.

Findings

Established a Galois correspondence between quantum function algebras and quantum universal enveloping algebras.

Provided explicit examples including semisimple groups, Euclidean group, and Heisenberg group.

Applied the framework to renormalization in quantum electrodynamics.

Abstract

The "quantum duality principle" states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie bialgebra as well. We extend this to a much more general result: namely, given any principal ideal domain R, for each prime h in R we establish sort of an "inner" Galois' correspondence on the category HA of torsionless Hopf algebras over R, via the definition of two functors (from HA to itself) such that the image of the first, resp. of the second, is the full subcategory of those Hopf algebras which are commutative, resp. cocommutative, modulo h (i.e. they are "quantum function algebras" (=QFA), resp. "quantum universal enveloping algebras" (=QUEA), at h). In particular we provide a machine to get two quantum groups - a QFA and a QUEA - out of any Hopf algebra H over a field k: just plug in a parameter x and apply the functors to H[x] for h = x. Several relevant examples are studied in full detail: the trivial quantisations, the semisimple groups, the Euclidean group, the Heisenberg group, and the Kostant-Kirillov structure on any Lie algebra; furthermore, an interesting application to renormalisation theory in quantum electro-dynamics is studied, as a sample of application of the principle to a quite large class of problems. This work is a far-reaching "evolution" of the same author's preprint math.QA/9912186: the present paper is entirely self-contained, is more general from the mathematical point of view, and contains additional examples. WARNING: This preprint has been overtaken by a new, deeply enhanced and improved version, available as {\tt math.QA/0303019}; the interested reader is kindly asked to refer to that new preprint.