The global quantum duality principle: theory, examples, and applications

Abstract

Let R be an integral domain, h non-zero in R such that R/hR is a field, and HA the category of torsionless (or flat) Hopf algebras over R. We call any H in HA "quantized function algebra" (=QFA), resp. "quantized (restricted) universal enveloping algebra" (=QrUEA), at h if 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. We establish an "inner" Galois' correspondence on HA, via the definition of two endofunctors, ()^\vee and ()', of HA such that: (a) the image of ()^\vee, resp. of ()', is the full subcategory of all QrUEAs, resp. all QFAs, at h; (b) if R/hR has zero characteristic, the restriction of ()^\vee to QFAs and of ()' to QrUEAs yield equivalences inverse to each other; (c) if R/hR has zero characteristic, starting from a QFA over a Poisson group, resp. from a QrUEA over a (restricted) Lie bialgebra, the functor ()^\vee, resp. ()', gives a QrUEA, resp. a QFA, over the dual Lie bialgebra, resp. the dual Poisson group. In particular, (a) yields a recipe to produce quantum groups of both types (QFAs or QrUEAs), (b) gives a characterization of them within HA, and (c) gives a "global" version of the "quantum duality principle" after Drinfeld. We then apply our result to Hopf algebras defined over a field k and extended to the polynomial ring k[h]: this yields quantum groups, hence "classical" geometrical symmetries of Poisson type (via specialization) associated to the "generalized symmetry" encoded by the original Hopf algebra over k. Both the main result and the above mentioned application are illustrated via several examples of many different kinds, which are studied in full detail.