The global quantum duality principle

Let R be an integral domain, let h in R be anon-zero element such that k := R/hR is a field, and let \HA be the category of torsionless (or flat) Hopf algebras over R. We call an object H in \HA a "quantized function algebra" (in short, a QFA), resp. "quantized restricted universal enveloping algebra" (in short, a QrUEA), at h if — roughly speaking — the quotient H/hH is the function algebra of a connected Poisson group, resp. the (restricted, if R/hR has positive characteristic) universal enveloping algebra of a (restricted) Lie bialgebra. Extending a result of Drinfeld, we establish an "inner" Galois' correspondence on \HA, via two endofunctors, ( )^\vee and ( )', of \HA such that H^\vee is a QrUEA and H' is a QFA (for all H in \HA). In addition: (a) the image of ( )^\vee, resp. of ( )', is the full subcategory of all QrUEAs, resp. of all QFAs; (b) if p := Char(k) = 0, the restrictions of ( )^\vee to QFAs and of ( )' to QrUEA yield equivalences inverse to each other; (c) if p=0, starting from a QFA over a Poisson group G, resp. from a QrUEA over a Lie bialgebra g, the functor ( )^\vee, resp. ( )', gives a QrUEA, resp. a QFA, over the dual Lie bialgebra, resp. the dual Poisson group. Several, far-reaching applications are developed in detail in [Ga2]–[Ga4].