Integrals in Hopf algebras over rings

• Type: Article in International Scientific Journal
• Date: 2004

Authors

Christian Lomp (Author) FCUP

Journal

Title: Communications in Algebra

Vol. 32 No. 12

Pages: 4687-4711

ISSN: 0092-7872

Publisher: Taylor & Francis

Indexing

ISI Web of Knowledge - 11 Citations

ISI Web of Science

Scopus - 10 Citations

Scientific classification

FOS: Natural sciences > Mathematics

Other information

Authenticus ID: P-000-CF9

DOI: 10.1081/agb-200036837

Resumo (PT): Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedler's equivalence is still valid for free Hopf algebras or projective Hopf algebras over integral domains. Analogously for a left H-module algebra A we study the influence of non-zero left A#H-linear maps from A to A#H on H being finitely generated as R-module. Examples and application to separability are given.

Abstract (EN): Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedler's equivalence is still valid for free Hopf algebras or projective Hopf algebras over integral domains. Analogously for a left H-module algebra A we study the influence of non-zero left A#H-linear maps from A to A#H on H being finitely generated as R-module. Examples and application to separability are given.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 25