Throughout, H is a Hopf algebra over a field k of characteristic p, G(H) is the group of grouplikes of H and L is any subgroup of G(H). We denote the antipode of H by S. We investigate the freeness of Hopf algebras as modules over their group algebras of grouplikes. / In chapter II we consider semisimple group algebras kL. We prove that for finite dimensional H all nonzero objects in the category (' )(, ) of left (H,kL)-Hopf modules are free kL-modules. We also prove this in the case when S('2) = id. Hence, for a finite dimensional H, the number of one-dimensional ideals divides the dimension of H and the order of S divides 4(.)dimension of H. / In chapter III we prove that a finite dimensional H is a free k-module for any g (ELEM) G(H), even if p divides the order of g. Further we establish that a finite dimensional H is a free kL-module if and only if H is a free kA-module for any elementary abelian p-subgroup A of L. / In chapter IV we prove that a finite dimensional H over an algebraically closed field k of characteristic p is a free kL-module, if H does not contain any simple subcoalgebra of dimension (lp)('2) for any natural number 1 (GREATERTHEQ) 2. Further we construct an example of an infinite dimensional H showing that not all objects in (' )(, ) are free kL-modules. Finally we show that any infinite dimensional H is a free kL-module, if L is an infinite group which contains no nontrivial finite subgroup. Also, if the dimension of H equals the dimension of the coradical of H, then H is a free module over any of its semisimple group algebras k where g (ELEM) G(H). / Source: Dissertation Abstracts International, Volume: 47-01, Section: B, page: 0246. / Thesis (Ph.D.)--The Florida State University, 1985.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_75766 |
| Contributors | ZOELLER, MARTHA BETTINA., Florida State University |
| Source Sets | Florida State University |
| Detected Language | English |
| Type | Text |
| Format | 91 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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