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Title: Unipotent subgroups of reductive algebraic groups
Author: Proud, Richard
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1997
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Let G be a connected reductive algebraic group defined over an algebraically closed field of good characteristic p>0. Suppose uEG has order p. In [T2] it is shown that u lies in a closed reductive subgroup of G of type Al. This is the best possible group theoretic analogue of the Jacobson-Morozov theorem for simple Lie algebras. Testerman's key result is a type of `exponentiation process'. For our given element u, this process constructs a 1-dimensional connected abelian unipotent subgroup of G, hence isomorphic to Ga, containing u. This in turn yields the required Al overgroup of u. Now let 1#uEG be an arbitrary unipotent element. Such an element has order pt, for some tEN. In this thesis we extend the above result, and show that u lies in a t-dimensional closed connected abelian unipotent subgroup of G, provided p> 29 when G' contains a simple component of type E8, and that p is good for the remaining components. The structure of the resulting unipotent overgroup is also explicitly given. This is the best possible result, in terms of `minimal dimension', which we could hope for. In Chapter 1 we discuss the theory of Witt vectors, associated with a commutative ring with identity. They are closely related to the study of connected abelian unipotent algebraic groups. The unipotent overgroups are constructed using a variation of the usual exponen- tiation process. The necessary material on formal power series rings is given in 1.3. The Artin-Hasse exponentials of 1.4 play a crucial role in this construction. The connection between Witt groups and Artin-Hasse exponentials is discussed in 1.5. In Chapter 2 we apply the techniques of Chapter 1 to the various simple algebraic groups. For each type, a particular isogeny class is chosen and the required overgroup is constructed for the regular (and subregular) classes. In 2.9 we pass to the adjoint case. In Chapter 3 we extend the results of Chapter 2 to include all unipotent classes in all reductive algebraic groups (under certain restrictions). In 3.1 the Cayley Transform for the classical groups is combined with the ideas of Chapter 1 to give an explicit construction of the unipotent overgroups for every unipotent class. In 3.2 we discuss semiregular unipotent elements. Finally, in 3.3, we prove the main theorem of this thesis.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council (EPSRC) ; Institute of Mathematics, University of Warwick
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics