Finite-Dimensional Simple-Pointed Hopf Algebras

Abstract

for some q g k _ 0. With the advent of quantum groups these relations took on added importance in the theory of Hopf algebras. The quantized w x enveloping algebras, see 2, 5 , for example, are generated by pairs of such elements a and x which satisfy these relations. Quantized enveloping algebras are examples of pointed Hopf algebras. A natural question to ask about pointed Hopf algebras is which ones are ‘‘simple’’ in an appropriate sense. In this paper we give a definition of ‘‘simple’’ pointed Hopf algebra and refer to such Hopf algebras as simplepointed. We describe the structure of simple-pointed Hopf algebras in the class of pointed Hopf algebras which are generated by pairs a and x which satisfy the relations above when k is algebraically closed. In the finite-dimensional characteristic 0 case we characterize the coalgebra structure of the duals of these simple-pointed Hopf algebras as well. Many finite-dimensional Hopf algebras A over a field k are non-trivial biproducts. A necessary and sufficient condition for A to be a biproduct is the existence of a Hopf algebra projection A a H from A onto a sub-Hopf algebra H of A. The associated biproduct realization of A has