Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9493589 | Journal of Algebra | 2005 | 39 Pages |
Abstract
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(gâ) over a Lie bialgebra gâ. 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(kâ)) for some Poisson group K+ and some Lie bialgebra kâ. 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 gâ (with g:=Lie(G)), and the first Poisson group associated to H=U(g) is of type Gâ, 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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Fabio Gavarini,