Unipotent subgroups of reductive algebraic groups

by Richard Proud

Publisher: typescript in [s.l.]

Written in English
Published: Pages: 138 Downloads: 585
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Edition Notes

Thesis (Ph.D.) - University of Warwick, 1997.

StatementRichard Proud.
The Physical Object
Paginationvi,138p.
Number of Pages138
ID Numbers
Open LibraryOL17524805M

As a corollary, we obtain a complete set of generic canonical representatives for the unipotent classes of the finite general unitary groups GUn(Fq) for all prime powers q. Our second topic is concerned with unipotent pieces, defined by G. Lusztig in [Unipotent elements in small characteristic, Transform. Groups 10 (), ]. The Structure of Complex Lie Groups addresses this need. Self-contained, it begins with general concepts introduced via an almost complex structure on a real Lie group. It then moves to the theory of representative functions of Lie groups- used as a primary tool in subsequent chapters-and discusses the extension problem of representations that. Classification of Pseudo-reductive Groups (AM) - Ebook written by Brian Conrad, Gopal Prasad. Read this book using Google Play Books app on your PC, android, iOS devices. Download for offline reading, highlight, bookmark or take notes while you read . Multiplicity-free subgroups of reductive algebraic groups Jonathan Brundan Abstract We introduce the notion of a multiplicity-free subgroup of a reductive alge-braic group in arbitrary characteristic. This concept already exists in the work of Kr amer for compact connected Lie groups. We give a .

NILPOTENT ELEMENTS AND REDUCTIVE SUBGROUPS OVER A LOCAL FIELD GEORGE J. MCNINCH Abstract. Let K be a local field – i.e. the field of fractions of a complete DVR A whose residu.   Linear Algebraic Groups by James E. Humphreys, Structure of Reductive Groups.- .- Standard Parabolic Subgroups.- Levi Decompositions.- Parabolic Subgroups Associated to Certain Unipotent Groups.- Maximal Subgroups and Maximal Unipotent Subgroups.- XI/5(9). hence non-a ne, algebraic groups. We now gather some basic properties of algebraic groups: Lemma Any algebraic group Gis a smooth variety, and its (connected or irreducible) com-ponents are the cosets gG 0, where g2G. Moreover, G is a closed normal . of abelian algebraic groups, and in particular abelian connected unipotent groups, shows that one should aim for a t-dimensional group if o(u) = pt. The rst work in this direction was done by Richard Proud in [Pro01], who showed: Theorem [Pro01, Main Theorem] Let Gbe a .

Let G be a reductive algebraic group over an algebraically closed field k (i.e. G has no normal unipotent subgroup of positive dimension). Note that G need not be connected and is reductive if and only if its connected component containing 1 is reductive. Conjugacy classes of unipotent elements in algebraic groups with simple con-. This book is based on the notes of the authors' seminar on algebraic and Lie groups held at the Department of Mechanics and Mathematics of Moscow University in / Our guiding idea was to present in the most economic way the theory of semisimple Lie groups on the basis of the theory of algebraic groups. Statement In characteristic zero: connected case. Suppose is an algebraically closed field of characteristic zero and is a finite-dimensional connected unipotent abelian algebraic group , is isomorphic, as an algebraic group, to the direct product of finitely many copies of the additive group r, the number of copies used equals the dimension of.

Unipotent subgroups of reductive algebraic groups by Richard Proud Download PDF EPUB FB2

ABELIAN UNIPOTENT SUBGROUPS OF REDUCTIVE GROUPS GEORGE J. MCNINCH ABSTRACT. Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold.

First, when p is a good prime, we give a new proof of the “order formula” of D. Tester. The study of algebraic groups, regarded as groups of matrices, is almost as old as group theory work on unipotent subgroups and his own work on the classification of simple algebraic reductive groups (over perfect fields, the two are the same).

This lacuna was filled by the. A question on a unipotent element in reductive algebraic groups. Ask Question Asked 9 years, 3 months ago. You can also see the book by Jantzen to find what Professor Premet said. The book is Nilpotent orbits in representation theory pagesin ''Lie Theory, Lie algebras and representations'', Progress in MathematicsBirhauser.

Abelian Unipotent Subgroups of Reductive Groups Article in Journal of Pure and Applied Algebra () February with 12 Reads How we measure 'reads'. Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications.

This self-contained monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a Cited by: Solvable algebraic groups are studied in detail in Chapters The final eight chapters treat the Borel-Chevalley structure theory of reductive algebraic groups over arbitrary fields.

Three appendices review the algebraic geometry needed, the construction of very general quotients of algebraic groups, and the theory of root data. Unipotent centralizers in algebraic groups.

Reductive Subgroups of Exceptional Algebraic Groups. generally and to establish a uniqueness result for the resulting unipotent subgroups. As a. Introduction. Let k be an algebraically closed field of characteristic p>0, and let G be a connected, reductive group over consider in this paper two questions which involve the relationship between nilpotent elements in the Lie algebra g of G and certain unipotent subgroups of G.

There are finitely many (adjoint) orbits of G on the nilpotent elements of its Lie algebra g; since Cited by: Buy Pseudo-reductive Groups (New Mathematical Monographs) on FREE SHIPPING on qualified orders Pseudo-reductive Groups (New Mathematical Monographs): Conrad, Brian, Gabber, Ofer, Prasad, Gopal: : Books.

Unipotent elements and parabolic subgroups of reductive groups. in Seminar in Algebraic groups and related fields, ed. Borel, Springer Lecture Notes About this paper. Cite this paper as: Tits J. () Unipotent elements and parabolic subgroups of reductive groups.

In: Cohen A.M., Hesselink W.H., van der Kallen W.L.J Cited by:   P. Mosch, On adjoint and coadjoint orbits of maximal unipotent subgroups of reductive algebraic groups, PhD Thesis, Ruhr-University Bochum, [30] R. Richardson, Conjugacy classes in Lie algebras and algebraic groups, Ann.

Math. 86 (), 1–Cited by: Unipotent Algebraic Groups (Lecture Notes in Mathematics) th Edition. by T. Kambayashi (Author), M. Miyanishi (Author), M. Takeuchi (Author) & ISBN ISBN Why is ISBN important. ISBN.

This bar-code number lets you verify that you're getting exactly the right version or edition of a book. Cited by: Here the setting is less clear, since "unipotent" typically comes up only for algebraic groups and the context is a local field (presumably not just the real field).

Indeed, the notion of "reductive" was first explored systematically by Borel and Tits, though the name is perhaps misleading in prime characteristic. Root groups in pseudo-reductive groups 67 Representability of automorphism functors 78 3 Basic structure theory 86 Perfect normal subgroups of pseudo-reductive groups 86 Root datum for pseudo-reductive groups 94 Unipotent groups associated to semigroups of roots 99 Bruhat decomposition and Levi subgroups Algebraic groups play much the same role for algebraists as Lie groups play for analysts.

This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in Cited by: Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications.

This self-contained monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification Price: $ Get this from a library. Unipotent and nilpotent classes in simple algebraic groups and Lie algebras.

[M W Liebeck; Gary M Seitz] -- "This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras.

These. The invariant theory of non-reductive groups has its roots in the 19th century but has seen some very interesting developments in the past twenty years. This book is an exposition of several related topics including observable subgroups, induced modules, maximal unipotent subgroups of reductive groups and the method of U-invariants, and the.

The next two results concern comparions between centralizers of unipotent elements in finte groups with centralizers in appropriate algebraic groups.

THEOREM - 5. Suppose p # 2 is a good prime, G finite, and G = OP’(GU) for d a quasisimple algebraic group over k and a an endomorphism of G. "Linearly reductive" and "reductive" are equivalent when the base field is of characteristic zero, but for prime characteristic they are different -- in fact, in prime characteristic, the only connected linearly reductive groups are algebraic tori, though for example $\operatorname{GL}_n$ is reductive.

Property 3. says that, up to the extension problem. all linear algebraic groups can be understood in terms of reductive and unipotent groups. Both type of groups are relatively simple—reductive groups have nice representation theory (and, in fact, a classification theorem!), and unipotent groups are just iterated extensions of.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up. Simple root subgroups of split reductive groups multiply nicely. When does the Borel subgroups of affine (linear) algebraic groups come from Borel subgroups of general.

Abstract: The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on the maximal unipotent subgroup of a split reductive group and show how this improves computation in the reductive group by: 2.

Algebraic Groups I. Unipotent radicals and reductivity In class, we have proved the important fact that over any eld k, a non-solvable connected reductive group containing a 1-dimensional split maximal k-torus is k-isomorphic to SL 2 or PGL 2.

That proof relied on knowing that maximal tori remain maximal after a ground eld extension to. Get this from a library. Unipotent and nilpotent classes in simple algebraic groups and Lie algebras.

[M W Liebeck; Gary M Seitz] -- This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in.

Unipotent and nilpotent classes in simple algebraic groups and lie algebras / Martin W. Liebeck, Gary M. Seitz. Labellings of some distinguished classes in classical groups Chapter Reductive subgroups Chapter Annihilator spaces of nilpotent elements This book concerns the theory of unipotent elements in simple.

Parabolic Subgroups.- Standard Parabolic Subgroups.- Levi Decompositions.- Parabolic Subgroups Associated to Certain Unipotent Groups.- Maximal Subgroups and Maximal Unipotent Subgroups.- XI. Representations and Classification of Semisimple Groups.- Price: $ Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications.

This monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form. We assume that the reader is familiar with the general theory of algebraic groups and the core theory of unipotent elements.

Nevertheless, in this introduction we shall review the latter, before outlining the content of the thesis. We shall also review more specialist background material as required. Basic facts about unipotent, nilpotent and. Algebraic groups play much the same role for algebraists as Lie groups play for analysts.

This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in. eld. This number is called the nilpotent rank of Gin SGA3, and the rank of Gin Borel’s book.

For example, if Gis reductive then it turns out (as is shown in the handout \Unipotent radicals and reductivity") that C= T. That is, in a connected reductive group the Cartan subgroups are .COVID campus closures: see options for getting or retaining Remote Access to subscribed contentCited by: 8.

In this first book, Marc Cabanes and Michel Enguehard introduce us to the study of the representations of a particular class of finite groups. These groups, which can be described as the groups you get by taking the points over a finite field of a reductive algebraic groups, include (in a sense) most of the finite simple groups.