Connected Hopf Algebras in Positive Characteristic

This talk is a combination of a general introduction to Hopf algebras from an algebraic point of view and my individual work and recent joint work with Linhong Wang on finite dimensional connected (pointed) Hopf algebras over an algebraically closed field of positive characteristic $p$. In the first part, I will provide background materials for a basic understanding of Hopf algebras. In particular, several open questions will be discussed concerning representations and structures of certain Hopf algebras.

In the second part, I will explain our results and methods for classification of connected Hopf algebras in low dimensions $p$, $p^2$ and $p^3$. This work is a completion of Henderson's classification in the graded co-commutative case and overlaps part of Scherotzke's classification work for rank one pointed Hopf algebras. Moreover, these results can be compared with the well-known classification work in characteristic zero for Hopf algebras of same dimensions, which were obtained by Andruskiewitsch-Schneider, Caenepeel-Dascalescu, Masuoka, Ng, Stefan-Van Oystaeyen, Zhu and others.