The global quantum duality principle

Abstract

Let R be an integral domain, let a non-zero h in R be such that k := R/hR is a field, and let HA be the category of torsionless (or flat) Hopf algebras over R. We call H in HA a "quantized function algebra" (=QFA), resp. "quantized restricted universal enveloping algebras" (=QrUEA), at h if - roughly speaking - 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(R/hR) = 0 , the restrictions of ( )^\vee to the QFAs and of ( )' to the QrUEAs 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 [Ga2-4].