Classification of Hermitean Forms in Characteristic 2

1979 ◽  
pp. 151-168
Author(s):  
Herbert Gross
Keyword(s):  
Characteristic 2

scholarly journals Algebraic Surfaces of General Type with Small c21 in Positive Characteristic

2008 ◽  
Vol 191 ◽  
pp. 111-134 ◽  
Author(s):  
Christian Liedtke
Keyword(s):  
Characteristic Zero ◽  
General Type ◽  
Positive Characteristic ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Arbitrary Characteristic ◽  
Numerical Invariants ◽  
Characteristic 2

AbstractWe establish Noether’s inequality for surfaces of general type in positive characteristic. Then we extend Enriques’ and Horikawa’s classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

scholarly journals Spherical subgroups in simple algebraic groups

2015 ◽  
Vol 151 (7) ◽  
pp. 1288-1308
Author(s):  
Friedrich Knop ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Closed Subgroup ◽  
Positive Characteristic ◽  
Dense Orbit ◽  
Simple Algebraic Group ◽  
Spherical Subgroup ◽  
Group A ◽  
General Deformation ◽  
Characteristic 2

Let $G$ be a simple algebraic group. A closed subgroup $H$ of $G$ is said to be spherical if it has a dense orbit on the flag variety $G/B$ of $G$. Reductive spherical subgroups of simple Lie groups were classified by Krämer in 1979. In 1997, Brundan showed that each example from Krämer’s list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, up to now there has been no classification of all such instances in positive characteristic. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic 2 which has no counterpart in Krämer’s classification. As one of our key tools, we prove a general deformation result for subgroup schemes that allows us to deduce the sphericality of subgroups in positive characteristic from the same property for subgroups in characteristic zero.

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.

Overview: The classification of groups of characteristic 2 type

2011 ◽  
pp. 63-81
Author(s):  
Michael Aschbacher ◽  
Richard Lyons ◽  
Stephen Smith ◽  
Ronald Solomon
Keyword(s):  
Characteristic 2

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.

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.

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