The crystal duality principle: from Hopf algebras to geometrical symmetries

We give functorial recipes to get, out of any Hopf algebra over a field, two pairs of Hopf algebras with some geometrical content. If the ground field has characteristic zero, the first pair is made by a function algebra F[G_+] over a connected Poisson group and a universal enveloping algebra U(g_−) over a Lie bialgebra g_− . In addition, the Poisson group as a variety is an affine space, and the Lie bialgebra as a Lie algebra is graded. Forgetting these last details, the second pair is of the same type, namely (F[K_+],U(k_−)) for some Poisson group K_+ and some Lie bialgebra k_− . When the Hopf algebra H we start from is already of geometric type the result involves Poisson duality. The first Lie bialgebra associated to H = F[G] is g∗ - with g := Lie(G) - and the first Poisson group associated to H = U(g) is of type G∗ , i.e. it has g as cotangent Lie bialgebra. If the ground field has positive characteristic, the same recipes give similar results, but the Poisson groups obtained have dimension 0 and height 1, and restricted universal enveloping algebras are obtained. We show how these geometrical Hopf algebras are linked to the initial one via 1-parameter deformations, and explain how these results follow from quantum group theory. We examine in detail the case of group algebras.