Finite-Dimensional Simple-Pointed Hopf Algebras

✍ Scribed by David E Radford

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 201 KB
• Volume: 211
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7613

✦ Synopsis

Some of the first examples of Hopf algebras described over a field k w x which are neither commutative nor cocommutative 13, 14 involve elements a and x which satisfy the relations ⌬ a s a m a, ⌬ x s x m a q 1 m x, and xa s qax Ž . Ž .

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