Hopf algebras of low dimension

Abstract

The main aim of this paper is to classify all types of Hopf algebras of dimension less thn or equal to 11 over an algebraically closed field of characteristic 0. IfAis such a Hopf algebra that is not semisimple, then we shall prove that A or A* is pointed. This property will result from the fact that, under some assumptions, any Hopf algebra that is generated as an algebra by a four-dimensional simple subcoalgebra is a Hopf quotient of the coordinate ring of quantum SL2(k). The first result allows us to reduce the classification to the case of pointed Hopf algebras of dimension 8. We shall describe their types in the last part of the paper. © 1999 Academic Press.