Affine commutative-by-finite Hopf algebras

Abstract

The objects of study in this paper are Hopf algebras H which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra A by a finite dimensional Hopf algebra H‾:=H/A+H, where A+ is the augmentation ideal of A. Basic structural and homological properties are recalled and classes of examples are listed. A structure theorem is proved when (⁎)H‾ is semisimple and cosemisimple, showing that in this case the noncommutativity of H arises from the action of a finite group. For example, when (⁎) holds and H is prime and pointed, it is a crossed product of a smooth affine commutative domain by a finite group, and the simple H-modules are described by a type of Clifford's theorem.