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The global quantum duality principle: a survey through examples

Let R be a 1-dimensional integral domain, let h (non-zero) be a prime element, and let \HA be the category of torsionless Hopf algebras over R. We call H in \HA a "quantized function algebra" (=QFA), resp. "quantized restricted universal enveloping algebras" (=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. An "inner" Galois correspondence on \HA is established 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. QFAs, at h; (b) if p := Char(R/hR) = 0, the restrictions of ( )^\vee to QFAs and of ( )' to QrUEAs yield equivalences inverse to each other; (c) if p = 0, then 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. a dual Poisson group. In particular, (a) provides a machine to produce quantum groups of both types (either QFAs or QrUEAs), (b) gives a characterization of them among objects of \HA, and (c) gives a "global" version of the so-called "quantum duality principle" (after Drinfeld's, cf. [Dr]). These notes draw a sketch of the construction leading to the "global quantum duality principle". Besides, the principle itself, and in particular the above mentioned application, is illustrated by means of several examples. There are no proofs, but all (of them, and any other detail) can be found instead in arXiv:math/0303019v8 [math.QA].