A characterization of quantum groups Hans-Jürgen Schneider

In this talk I will report on joint work with N. Andruskiewitsch on the classification of certain classes of Hopf algebras over an algebraically closed field of characteristic zero.

A Hopf algebra A is pointed if all its simple comodules are one-dimensional. Let grA be the graded Hopf algebra associated to the coradical filtration of A. The degree zero part of grA is the coradical A_{0} of A. A_{0} is the group algebra of the group of group-like elements of A which we call Γ. Let R be the subalgebra of A_{0}-coinvariant elements. The primitive elements P (R) of R form a Yetter-Drinfeld module over Γ.

It turns out that the braiding of P (R) is a crucial invariant of A. We call it the infinitesimal braiding of A. We call the braiding positive if it is given by a matrix with positive diagonal entries ≠ 1.

In the main Theorem 5.2 of [AS5] we classify all Hopf algebras which are • pointed and have an abelian group of group-like elements Γ,

• domains of finite Gelfand-Kirillov dimension, and

• have positive infinitesimal braiding .

In Theorem 4.2, we describe these Hopf algebras by generators and relations. They are natural generalizations of quantized enveloping algebras with positive parameter.

The proofs in [AS5] rely on work of Rosso [Ro] and on our previous papers [AS1],[AS2],[AS3].

In [AS3] we classify all finite-dimensional pointed Hopf algebras with Γ ≅ (ℤ/(p))^{s}, p a prime > 17.

[AS4] is a survey paper on our project, and it contains the classification of all finite-dimensional pointed Hopf algebras with infinitesimal braiding of type A_{n}.

References

[AS1] N. Andruskiewitsch and H.-J. Schneider, Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p^{3}, J. Algebra 209 (1998), 658–691.

[AS2] , Finite quantum groups and Cartan matrices, Adv. in Math. 154 (2000), 1–45.

[AS3] , Finite quantum groups over abelian groups of prime exponent, Ann. Sci. Ec. Norm. Super., to appear.

[AS4] , Pointed Hopf Algebras, in “Recent developments in Hopf algebra Theory”, MSRI series Cambridge Univ. Press; to appear.

[AS5] , A characterization of quantum groups, math.QA/0201095.

[Ro] M. Rosso, Quantum groups and quantum shuffles, Inventiones Math. 133 (1998), 399–416.

Mathematisches Institut, Universität München, Theresienstraße 39, D-80333 München, Germany

   E-mail address: hanssch@rz.mathematik.uni-muenchen.de
   URL: http://www.mathematik.uni-muenchen.de/∼hanssch/