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Saturday, July 18, 2020 | History

4 edition of A₁ subgroups of exceptional algebraic groups found in the catalog.

# A₁ subgroups of exceptional algebraic groups

## by R. Lawther

• 142 Want to read
• 11 Currently reading

Published by American Mathematical Society in Providence, R.I .
Written in English

Subjects:
• Linear algebraic groups.,
• Lie algebras.

• Edition Notes

Classifications The Physical Object Statement R. Lawther, D.M. Testerman. Series Memoirs of the American Mathematical Society,, no. 674 Contributions Testerman, Donna M., 1960- LC Classifications QA3 .A57 no.674, QA179 .A57 no.674 Pagination viii, 131 p. ; Number of Pages 131 Open Library OL38728M ISBN 10 0821819666 LC Control Number 99027224

rem that a simple algebraic group over an algebraically closed field has only finitely many conjugacy classes of maximal subgroups of positive dimension. It also follows that the maximal subgroups of sufficiently large order in finite exceptional groups of Lie type are known. Received by . Included is a brief discussion of modules occurring within parabolic subgroups and connections with unipotent classes. The section concludes with a discussion of the determination of the maximal closed connected subgroups of simple algebraic groups. The final section is an overview of the representation theory of simple algebraic groups.

A large part of the research in this area in the past two decades has been motivated by the potential application of algebraic group properties to the study of the finite simple groups of Lie type. The main result of this paper is a contribution to the study of positive dimensional closed subgroups of the classical algebraic by: 8. Memoirs of the American Mathematical Society, , Year: Cited by: 2.

Let G be a simple algebraic group over an algebraically closed field. A closed subgroup H of G is called G-completely reducible (G-cr) if, whenever H is contained in a parabolic subgroup P of G, it is contained in a Levi factor of P. In this paper we complete the classification of connected G-cr subgroups when G has exceptional type, by determining the L₀-irreducible connected reductive Author: Alastair J. Litterick, Alastair J. Litterick, Adam R. Thomas. [22], The maximal subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 90 (), no. The maximal subgroups of positive dimension in exceptional algebraic groups Jan

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### A₁ subgroups of exceptional algebraic groups by R. Lawther Download PDF EPUB FB2

Get this from a library. A₁ subgroups of exceptional algebraic groups. [R Lawther; Donna M Testerman] Add tags for "A₁ subgroups of exceptional algebraic groups". Be the first. Similar Items.

Related Subjects: (5) Book\/a>, schema. Electronic books: Additional Physical Format: Print version: Lawther, R. A₁ subgroups of exceptional algebraic groups / Material Type: Document, Internet resource: Document Type: Internet Resource, Computer File: All Authors / Contributors: R Lawther; Donna M Testerman.

Let $$G$$ be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $$p$$. Under some mild restrictions on $$p$$, we classify all conjugacy classes of closed connected subgroups $$X$$ of type $$A_1$$; for each such class of subgroups, we also determine the connected centralizer and the composition factors in the action on the Lie algebra $${\mathcal L}(G)$$.

In this article we present some recent results concerning the subgroups of the simple algebraic groups of exceptional type, and of the corresponding finite groups of Lie type. There are six sections. The first contains some general observations, while in the second we focus on connected subgroups.

The third section contains results on infinite closed subgroups, and in the last three sections we discuss finite by: 8. Maximal Subgroups of Exceptional Algebraic Groups Base Product Code Keyword List: memo ; MEMO ; memo/90 ; MEMO/90 ; memo ; MEMO ; memo/90/ ; MEMO/90/ ; memo ; MEMO Online Product Code: MEMO/90/E.

System Upgrade on Feb 12th During this period, E-commerce and registration of new users may not be available for up to 12 hours. For online purchase, please visit us again. A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup.

In this paper we classify all irreducible A 1 subgroups of exceptional algebraic groups uences are given concerning the representations of such subgroups on various G-modules: for example, the conjugacy classes of irreducible A 1 subgroups are determined by their Cited by: 4.

Maximal subgroups of exceptional algebraic groups The maximal subgroups M of positive dimension in exceptional algebraic groups have been completely classi ed by Liebeck and Seitz. They are maximal parabolics, maximal-rank subgroups, (22 D 4):Sym 3 5), or M0 is one of a short list: G M0 G 2 A 1 (p 7) F 4 A 1 (p.

Maximal subgroups of exceptional algebraic groups Let G be a simple algebraic group of exceptional type, deﬁned over an algebraically closed ﬁeld k of characteristic p ≥ 0.

Let M ⊂ G be a maximal positive-dimensional closed subgroup. By the Borel-Tits theorem, if M is not reductive, then M is a maximal parabolic subgroup of G. Let G be a simple algebraic group over an algebraically closed field.

A closed subgroup H of G is called G-completely reducible (G-cr) if, whenever H is contained in a parabolic subgroup P of G, it is contained in a Levi factor of this paper we complete the classification of connected G-cr subgroups when G has exceptional type, by determining the L 0-irreducible connected reductive Author: Alastair J.

Litterick, Alastair J. Litterick, Adam R. Thomas. subgroups H, and in Sections 3 and 4 for ﬁnite subgroups. A consequence of the proof of Theorem 1 is the following result concern-ing irreducible subgroups of ﬁnite exceptional groups. Corollary 2 Let σ be a Frobenius morphism of the exceptional algebraic group G, so that the ﬁxed point group Gσ is a ﬁnite exceptional group of Lie.

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irre Cited by: 4. The theory of simple algebraic groups is important in many areas of mathematics.

The authors of this book investigate the subgroups of certain types of simple algebraic groups and obtain a complete description of all those subgroups which are themselves simple. COVID Resources.

Reliable information about the coronavirus (COVID) is available from the World Health Organization (current situation, international travel).Numerous and frequently-updated resource results are available from this ’s WebJunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus.

The study of finite subgroups of a simple algebraic group $$G$$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $$G$$ has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of $$G$$, then the subgroup is called paper considers non-generic subgroups of simple algebraic groups of exceptional.

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible subgroups of exceptional algebraic. Algebraic Groups The theory of group schemes of ﬁnite type over a ﬁeld.

J.S. Milne Version Decem This is a rough preliminary version of the book published by CUP inThe final version is substantially rewritten, and the numbering has changed.

Let G be a simple algebraic group over an algebraically closed field k and let C 1,C t be non-central conjugacy classes in this paper, we consider the problem of determining whether there exist g i ∈ C i such that 〈 g 1,g t 〉 is Zariski dense in we establish a general result, which shows that if Ω is an irreducible subvariety of G t, then the set of tuples in Ω.

Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. 1 The Maximal Subgroups of Positive Dimension in Exceptional Algebraic Groups. The first eight chapters of the book study general algebraic group schemes over a field.

They culminate in a proof of the Barsotti-Chevalley theorem stating that every algebraic group is an extension of an abelian variety by an affine algebraic group. The remaining chapters treat only affine algebraic groups. An equivalent definition of a simple Lie group follows from the Lie correspondence: a connected Lie group is simple if its Lie algebra is a simple.

An important technical point is that a simple Lie group may contain discrete normal subgroups, hence being a simple Lie group is different from being simple as an abstract group.A number of consequences are obtained.

It follows from the main theorem that a simple algebraic group over an algebraically closed field has only finitely many conjugacy classes of maximal subgroups of positive dimension. It also follows that the maximal subgroups of sufficiently large order in finite exceptional groups of Lie type are known.In this book we will only consider the algebraic groups whose underlying varieties are affine ones.

They are called “affine” or “linear” algebraic groups. The difference between arbitrary groups and affine ones is quite essential from the point of view of algebraic geometry and almost indiscernible from the group-theoretical points of.