The global quantum duality principle

Abstract Let R be an integral domain, let ħ ∈ R\{0} be such that 𝕂 := R/ħR is a field, and let ℋ𝒜 be the category of torsionless (or flat) Hopf algebras over R. We call H ∈ ℋ𝒜 a ‘quantized function algebra’ (= QFA), resp. ‘quantized restricted universal enveloping algebra’ (= QrUEA), at ħ if—roughly speaking—H/ħH is the function algebra of a connected Poisson group, resp. the (restricted, if R/ħR 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, ( )∨ and ( )′, of ℋ𝒜 such that H ∨ is a QrUEA and H′ is a QFA (for all H ∈ ℋ𝒜). In addition: (a) the image of ( )∨, resp. of ( )′, is the full subcategory of all QrUEAs, resp. of all QFAs; (b) if p := Char(𝕂) = 0, the restrictions ( )∨|QFAs and ( )′|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 ( )∨, 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 [F. Gavarini, The global quantum duality principle: theory, examples, and applications, preprint 2003, http://arxiv.org/abs/math.QA/0303019] and [F. Gavarini, The Crystal Duality Principle: from Hopf Algebras to Geometrical Symmetries, J. Algebra 285 (2005), 399–437] and [F. Gavarini, Poisson geometrical symmetries associated to non-commutative formal diffeomorphisms, Commun. Math. Phys. 253 (2005), 121–155].