The Crystal Duality Principle: from Hopf Algebras to Geometrical Symmetries

Details The Crystal Duality Principle: from Hopf Algebras to Geometrical Symmetries The Crystal Duality Principle: from Hopf Algebras to Geometrical Symmetries

2003-04-14

arxiv.org/abs/math/0304164v4

Journal of Algebra 285 (2005), 399-437

Mathematics

Quantum Algebra

37 pages, AMS-TeX file. Minor typoes have been corrected in pages 23, 24, 26, 32 and 36. Final version, to appear in Journal o

Scientific paper

10.1016/j.jalgebra.2004.12.003

We give functorial recipes to get, out of any Hopf algebra over a field, two pairs of Hopf algebras which have some geometrical content. When the ground field has characteristic zero, the first pair is made by a function algebra over a connected Poisson group and a universal enveloping algebra over a Lie bialgebra. 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. 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 H = U(g) is of type G^*, i.e. it has g as cotangent Lie bialgebra. If the ground field has positive characteristic, then the same recipes give similar results, but for the fact that 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 original one via 1-parameter deformations, and explain how these results follow from quantum group theory. The cases of hyperalgebras and group algebras are examined in some detail (thus recovering some well-known, classical construction), along with some relevant examples.

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