Introduction

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Quadratic Forms ◽  
General Structure ◽  
Root Systems ◽  
Simple Groups ◽  
Reductive Groups ◽  
Central Extensions ◽  
New Techniques ◽  
Characteristic 2 ◽  
Minimal Type

This book deals with the classification of pseudo-reductive groups. Using new techniques and constructions, it addresses a number of questions; for example, whether there are versions of the Isomorphism and Isogeny Theorems for pseudosplit pseudo-reductive groups and of the Existence Theorem for pseudosplit pseudo-simple groups; whether the automorphism functor of a pseudo-semisimple group is representable; or whether there is a Tits-style classification in the pseudo-semisimple case recovering the version due to Tits in the semisimple case. This introduction discusses the special challenges of characteristic 2 as well as root systems, exotic groups and degenerate quadratic forms, and tame central extensions. It also reviews generalized standard groups, minimal type and general structure theorem, and Galois-twisted forms and Tits classification.

Classification of Pseudo-reductive Groups (AM-191)

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Quadratic Forms ◽  
General Structure ◽  
Arbitrary Field ◽  
Reductive Groups ◽  
Characteristic P ◽  
Central Extensions ◽  
New Techniques ◽  
Standard Construction ◽  
New Phenomena

This book goes further than the exploration of the general structure of pseudo-reductive groups to study the classification over an arbitrary field. An Isomorphism Theorem proved here determines the automorphism schemes of these groups. The book also gives a Tits-Witt type classification of isotropic groups and displays a cohomological obstruction to the existence of pseudo-split forms. Constructions based on regular degenerate quadratic forms and new techniques with central extensions provide insight into new phenomena in characteristic 2, which also leads to simplifications of the earlier work. A generalized standard construction is shown to account for all possibilities up to mild central extensions. The results and methods developed in this book will interest mathematicians and graduate students who work with algebraic groups in number theory and algebraic geometry in positive characteristic.

Central extensions and groups locally of minimal type

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Root System ◽  
Reductive Group ◽  
Reductive Groups ◽  
Central Extensions ◽  
General Lemma ◽  
Quotient Maps ◽  
Central Quotient ◽  
Characteristic 2 ◽  
Minimal Type ◽  
Quadratic Case

This chapter deals with central extensions and groups locally of minimal type. It begins with a discussion of the general lemma on the behavior of the scheme-theoretic center with respect to the formation of central quotient maps between pseudo-reductive groups; this lemma generalizes a familiar fact in the connected reductive case. The chapter then considers four phenomena that go beyond the quadratic case, along with a pseudo-reductive group of minimal type that is locally of minimal type. It shows that the pseudo-split absolutely pseudo-simple k-groups of minimal type with a non-reduced root system are classified over any imperfect field of characteristic 2. In this classification there is no effect if the “minimal type” hypothesis is relaxed to “locally of minimal type.”

Field-theoretic and linear-algebraic invariants

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Simple Group ◽  
Root System ◽  
Reductive Group ◽  
Simple Groups ◽  
Arbitrary Field ◽  
Isomorphism Classes ◽  
Algebraic Invariants ◽  
Characteristic 2 ◽  
Minimal Type

This chapter deals with field-theoretic and linear-algebraic invariants. It first presents a construction of non-standard pseudo-split absolutely pseudosimple k-groups with root system A1 over any imperfect field k of characteristic 2. It then considers an absolutely pseudo-simple group over a field k, along with a pseudo-split pseudo-reductive group over an arbitrary field k. It also establishes the equality over k of minimal fields of definition for projection onto maximal geometric adjoint semisimple quotients. This is followed by two examples that illustrate the root field in A1-cases. The chapter concludes with a discussion of a classification of the isomorphism classes of pseudo-split pseudo-simple groups G over an imperfect field k of characteristic p subject to the hypothesis that G is of minimal type. The associated irreducible root datum, which is sufficient to classify isomorphism classes in the semisimple case, is supplemented with additional field-theoretic and linear-algebraic data.

scholarly journals Classification of Spherical Fusion Categories of Frobenius–Schur Exponent 2

2021 ◽  
Vol 28 (01) ◽  
pp. 39-50
Author(s):  
Zheyan Wan ◽  
Yilong Wang
Keyword(s):  
Quadratic Forms ◽  
Gauss Sum ◽  
New Approach ◽  
Fusion Categories ◽  
Characteristic 2

In this paper, we propose a new approach towards the classification of spherical fusion categories by their Frobenius–Schur exponents. We classify spherical fusion categories of Frobenius–Schur exponent 2 up to monoidal equivalence. We also classify modular categories of Frobenius–Schur exponent 2 up to braided monoidal equivalence. It turns out that the Gauss sum is a complete invariant for modular categories of Frobenius–Schur exponent 2. This result can be viewed as a categorical analog of Arf's theorem on the classification of non-degenerate quadratic forms over fields of characteristic 2.

Universal smooth k-tame central extension

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Finite Type ◽  
Central Extension ◽  
Group Scheme ◽  
Reductive Groups ◽  
Central Extensions ◽  
Weil Restriction ◽  
Group A ◽  
Subgroup Scheme ◽  
Minimal Type ◽  
Simply Connected

This chapter describes the construction of canonical central extensions that are analogues for perfect smooth connected affine k-groups of the simply connected central cover of a connected semisimple k-group. A commutative affine k-group scheme of finite type is k-tame if it does not contain a nontrivial unipotent k-subgroup scheme. The chapter establishes good properties of the universal smooth k-tame central extension, noting that the property “locally of minimal type” is inherited by pseudo-reductive central quotients of pseudo-reductive groups. Although inseparable Weil restriction does not generally preserve perfectness, the chapter shows that the formation of the universal smooth k-tame central extension interacts with derived groups of Weil restrictions.

scholarly journals Finiteness theorems for algebraic groups over function fields

2011 ◽  
Vol 148 (2) ◽  
pp. 555-639 ◽  
Author(s):  
Brian Conrad
Keyword(s):  
Function Fields ◽  
Simple Groups ◽  
Structure Theory ◽  
Reductive Groups ◽  
Compactness Criterion ◽  
Group Schemes ◽  
Global Function Fields ◽  
Finiteness Theorems ◽  
Tamagawa Numbers

AbstractWe prove the finiteness of class numbers and Tate–Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and Godement’s compactness criterion (and a local analogue) for all such groups that are smooth and connected. This builds on the known cases of solvable and semi-simple groups via systematic use of the recently developed structure theory and classification of pseudo-reductive groups.

scholarly journals Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups

2016 ◽  
Vol 68 (2) ◽  
pp. 395-421 ◽  
Author(s):  
Skip Garibaldi ◽  
Daniel K. Nakano
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Quadratic Forms ◽  
Reductive Groups ◽  
Compact Lie Groups ◽  
Tilting Modules ◽  
Weyl Modules ◽  
Semisimple Complex ◽  
Characteristic 2 ◽  
Semisimple Algebraic Groups

AbstractThe representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the questions of whether a given complex representation is symplectic or orthogonal have been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from z hold by using similar proofs. This paper considers analogues of these results for simple, induced, and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.

scholarly journals Classification of Quadratic Forms over Skew Fields of Characteristic 2

2001 ◽  
Vol 240 (1) ◽  
pp. 366-392 ◽  
Author(s):  
Mohamed Abdou Elomary ◽  
Jean-Pierre Tignol
Keyword(s):  
Quadratic Forms ◽  
Skew Fields ◽  
Characteristic 2

Fusion of 2-elements in groups of finite Morley rank

2001 ◽  
Vol 66 (2) ◽  
pp. 722-730
Author(s):  
Luis-Jaime Corredor
Keyword(s):  
Algebraic Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Morley Rank ◽  
Finite Morley Rank ◽  
Abelian Subgroups ◽  
Characteristic 2

The Alperin-Goldschmidt Fusion Theorem [1, 5], when combined with pushing up [7], was a useful tool in the classification of the finite simple groups. Similar theorems are needed in the study of simple groups of finite Morley rank, in the even type case (that is, when the Sylow 2-subgroups are of bounded exponent, as in algebraic groups over fields of characteristic 2). In that context a body of results relating to fusion of 2-elements and the structure of 2-local subgroups is needed: pushing up, and the classification of groups with strongly or weakly embedded subgroups, or have strongly closed abelian subgroups (c.f, [2]). Two theorems of Alperin-Goldschmidt type are proved here. Both are needed in applications.The following is an exact analog of the Alperin-Goldschmidt Fusion Theorem for groups of finite Morley rank, in the case of 2-elements:Theorem 1.1. Let G be a group of finite Morley rank, and P a Sylow 2-subgroup of G. If A, B ⊆ P are conjugate in G, then there are subgroups Hi ≤ Pand elementsxi ∈ N(Hi) for 1 ≤ i ≤ n, and an elementy ∈ N(P), such that for all i:1. Hi is a tame intersection of two Sylow 2-subgroups;2. CP(Hi) ≤ Hi;3. N(Hi)/Hiis 2-isolatedand(a) (b) .

Preliminary notions

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Root System ◽  
Maximal Torus ◽  
Reductive Group ◽  
Root Systems ◽  
Positive System ◽  
Reductive Groups ◽  
Dynkin Diagrams ◽  
Standard Construction ◽  
Minimal Type ◽  
Split Maximal Torus

This chapter considers some preliminary notions, starting with standard pseudo-reductive groups, Levi subgroups, and root systems. It reviews the “standard construction” of pseudo-reductive k-groups and shows that any connected reductive group equipped with a chosen split maximal torus is generated by that maximal torus and its root groups for the simple positive and negative roots relative to a choice of positive system of roots in the root system. It also discusses the basic exotic construction, noting that the only nontrivial multiplicities that occur for the edges of Dynkin diagrams of reduced irreducible root systems are 2 and 3. Finally, it explains the minimal type pseudo-reductive k-group G, along with quotient homomorphism between pseudo-reductive groups.

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