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oai:www.ideals.illinois.edu:2142/22442 2023-07-10 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Haboush, William J. Wenzel, Christian 2011-05-07T13:39:57Z 2011-05-07T13:39:57Z 10000-01-01 1990 Given a semisimple linear algebraic group G over an algebraically closed field K, we fix a Borel subgroup B and a maximal torus T. This determines a root system $\Phi$, and a set of simple roots $\Delta$. The subgroups containing B are called parabolic subgroups. They correspond to subsets of $\Delta$. Thus there are finitely many. In this classical context, parabolic subgrous are understood to be varieties. In my thesis I generalize to subgroup-schemes containing B. They are group-schemes, but not necessarily varieties; their algebras of functions might have nilpotent elements, i.e. they might not be reduced. In my thesis I show that in characteristic p $>$ 0, there are infinitely many whenever G $\not=$ 1, I exhibit their structure, and I classify them. I show that in characteristic p $>$ 3, the subgroup-schemes containing B correspond to $\tilde\Delta$, the set of all maps from $\Delta$ to $\rm I\!N \cup \{\infty\},$ in such a way that it extends the classical classification of parabolic subgroups in terms of subsets of $\Delta$. To each $\varphi$ there is a parabolic P$\sb\varphi$ with $\rm P\sb\varphi = U\sb\varphi\cdot P\sb{I(\varphi)},$ I$(\varphi) = \{\alpha\in\Delta\mid\varphi(\alpha)=\infty\}$, $\rm P\sb{I(\varphi)} = (P\sb\varphi)\sb{red}$ = Spec(K (P$\sb\varphi$) /nilrad), U$\sb\varphi$ being a certain local unipotent subgroup-scheme. In characteristic 2,3 the situation is more complicated. Furthermore I give a construction of G/P also for non-reduced P. I show that G/P is a rational projective variety, whenever char (K) $>$ 3. Made available in DSpace on 2011-05-07T13:39:57Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114457.pdf: 2003410 bytes, checksum: 2508c8e7f65c047fb5ab06fb44c8fb0f (MD5) Previous issue date: 1990 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:57:39Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:27:03-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9114457 (UMI)AAI9114457 http://hdl.handle.net/2142/22442 eng Copyright 1990 Wenzel, Christian Mathematics Classification of all parabolic subgroup schemes of a semisimple linear algebraic group over an algebraically closed field of positive characteristic text Dissertation Ph.D. Mathematics Mathematics University of Illinois at Urbana-Champaign