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Publication# Special Reductive Groups Over An Arbitrary Field

Abstract

A linear algebraic group G defined over a field k is called special if every G-torsor over every field extension of k is trivial. In 1958 Grothendieck classified special groups in the case where the base field is algebraically closed. In this paper we describe the derived subgroup and the coradical of a special reductive group over an arbitrary field k. We also classify special semisimple groups, special reductive groups of inner type, and special quasisplit reductive groups over an arbitrary field k. Finally, we give an application to a conjecture of Serre.

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Related concepts (13)

Field extension

In mathematics, particularly in algebra, a field extension is a pair of fields K\subseteq L, such that the operations of K are those of L restricted to K. In this case, L is an extension

Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisel

Reductive group

In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a represen

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This dissertation is concerned with the study of irreducible embeddings of simple algebraic groups of exceptional type. It is motivated by the role of such embeddings in the study of positive dimensional closed subgroups of classical algebraic groups. The classification of the maximal closed connected subgroups of simple algebraic groups was carried out by E. B. Dynkin, G. M. Seitz and D. M. Testerman. Their analysis for the classical groups was based primarily on a striking result: if G is a simple algebraic group and ø : G → SL(V ) is a tensor indecomposable irreducible rational representation then, with specified exceptions, the image of G is maximal among closed connected subgroups of one of the classical groups SL(V), Sp(V ) or SO(V ). In the case of closed, not necessarily connected, subgroups of the classical groups, one is interested in considering irreducible embeddings of simple algebraic groups and their automorphism groups: given a simple algebraic group Y defined over an algebraically closed field K, one is led to study the embeddings G < Aut(Y ), where G and Aut(Y ) are closed subgroups of SL(V ) and V is an irreducible rational KY -module on which G acts irreducibly. A partial analysis of such embeddings in the case of classical algebraic groups Y was carried out by B. Ford. We purpose to classify all triples (G, Y, V ) where Y is a simple algebraic group of exceptional type, defined over an algebraically closed field K of characteristic p > 0, G is a closed non-connected positive dimensional subgroup of Y and V is a nontrivial irreducible rational KY -module such that V|G is irreducible. We obtain a precise description of such triples (G, Y, V ).

Fix an algebraically closed field $K$ having characteristic $p\geq 0$ and let $Y$ be a simple algebraic group of classical type over $K.$ Also let $X$ be maximal among closed connected subgroups of $Y$ and consider a non-trivial $p$-restricted irreducible rational $KY$-module $V.$ In this thesis, we investigate the triples $(Y,X,V)$ such that $X$ acts with exactly two composition factors on $V$ and see how it generalizes a question initially investigated by Dynkin in the $1950$s and then studied by numerous mathematicians. In particular, we study the natural embeddings of $\mbox{SO}_{2n}(K)$ in both $\mbox{Spin}_{2n+1}(K)$ and $\mbox{SL}_{2n}(K)$ and obtain results on the structure of certain Weyl modules.

We study (connected) reductive subgroups G of a reductive algebraic group H, where G contains a regular unipotent element of H. The main result states that G cannot lie in a proper parabolic subgroup of H. This result is new even in the classical case H = SL(n, F), the special linear group over an algebraically closed field, where a regular unipotent element is one whose Jordan normal form consists of a single block. In previous work, Saxl and Seitz (1997) determined the maximal closed positive-dimensional (not necessarily connected) subgroups of simple algebraic groups containing regular unipotent elements. Combining their work with our main result, we classify all reductive subgroups of a simple algebraic group H which contain a regular unipotent element.

2013