Throughout, H is a finite dimensional Hopf algebra with antipode S over an algebraically closed field k satisfying (dimH)1 $\ne$ 0. G(H) denotes the group of grouplikes. A Hopf algebra is said to be involutory if the square of its antipode is the identity. We study the mathematics related to Kaplansky's 5$\sp{th}$ conjecture, (1), that if the Hopf algebra H or its dual H* is semisimple as an algebra, then H is involutory. / Suppose that B is a sub Hopf algebra of H. We investigate the question under which assumptions B involutory implies H involutory. We succeed in showing that if H is cosemisimple and $rank\sb{B}H \leq$ 4, then H is semisimple and cosemisimple and $S\sp2$ = id. / If char(k) $\not=$ 2, we establish a relationship between $S\sp2$ and grouplikes $g\in G$(H) acting on simple subcoalgebras. It is shown that if A is a simple subcoalgebra A of dimension $n\sp2$ and $g\in$ G(H) is a grouplike of order n such that gA = A then $S\sp2\vert\sb{A} = id\sb{A}$. This enables us to verify Kaplansky's conjecture for cosemisimple Hopf algebras all of whose simple subcoalgebras have dimension 1 or $q\sp2$ with q prime. / The main part of our thesis concentrates on cosemisimple Hopf algebras all of whose simple subcoalgebras have dimension at most 9. It is shown that if A is a simple subcoalgebra A of dimension 9 and char(k) $\not=$ 2,3, then $S\sp2\vert\sb{A}$ has order 1, 2 or 3. If the characteristic of the field k is p $>$ dimH, then H is semisimple and cosemisimple. As a consequence, we are able to verify Kaplansky's conjecture for cosemisimple Hopf algebras with "small" simple subcoalgebras, if the characteristic of the field k is p $>$ (dimH)$\sp2$. / Source: Dissertation Abstracts International, Volume: 53-07, Section: B, page: 3520. / Major Professor: Warren D. Nichols. / Thesis (Ph.D.)--The Florida State University, 1992.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_76702 |
| Contributors | Eberwein, Martin., Florida State University |
| Source Sets | Florida State University |
| Language | English |
| Detected Language | English |
| Type | Text |
| Format | 107 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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