A Global Version of the Quantum Duality Principle

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, for any principal ideal domain R and for each prime p ∈ R we establish an “inner” Galois’ correspondence on the category HA of torsionless Hopf algebras over R, using two functors (from HA to itself) such that the image of the first and the second is the full subcategory of those Hopf algebras which are commutative and cocommutative, modulo p, respectively (i.e., they are “quantum function algebras” (=QFA) and “quantum universal enveloping algebras” (=QUEA), at p, respectively). 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: apply the functors to k[ν] ⊗k H for p = ν . A relevant example occurring in quantum electro-dynamics is studied in some detail.