Affine commutative-by-finite Hopf algebras

• Published in: Journal of algebra Vol. 573; pp. 56 - 94
• Main Authors: Brown, K.A., Couto, M.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-05-2021
• Subjects: Hopf algebra; Polynomial identity; secondary; Hopf algebra; Polynomial identity; primary
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/j.jalgebra.2020.12.039

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.