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.