Cubic Action of a Rank One Group.

Format:

Book

Author/Creator:

Grüninger, Matthias.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.276

Language:

English

Subjects (All):

Group theory.

Physical Description:

1 online resource (154 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"We consider a rank one group G = A,B acting cubically on a module V , this means [V, A, A,A] = 0 but [V, G, G,G] = 0. We have to distinguish whether the group A0 := CA([V,A]) CA(V/CV (A)) is trivial or not. We show that if A0 is trivial, G is a rank one group associated to a quadratic Jordan division algebra. If A0 is not trivial (which is always the case if A is not abelian), then A0 defines a subgroup G0 of G acting quadratically on V . We will call G0 the quadratic kernel of G. By a result of Timmesfeld we have G0 = SL2(J,R) for a ring R and a special quadratic Jordan division algebra J R. We show that J is either a Jordan algebra contained in a commutative field or a Hermitian Jordan algebra. In the second case G is the special unitary group of a pseudo-quadratic form of Witt index 1, in the first case G is the rank one group for a Freudenthal triple system. These results imply that if (V,G) is a quadratic pair such that no two distinct root groups commute and charV = 2, 3, then G is a unitary group or an exceptional algebraic group"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

Chapter 2. Preliminaries

2.1. Moufang sets

2.2. Rank one groups

2.3. Some ring theory

2.4. Jordan algebras

2.5. Envelopes of special Jordan algebras

2.6. Quadratic spaces and Clifford Jordan algebras

2.7. Involutory sets and pseudo-quadratic forms

2.8. Cubic norm structures

2.9. Freudenthal triple systems

2.10. Structurable algebras

2.11. The Clifford algebra of a Freudenthal triple system

Chapter 3. Cubic Action

Chapter 4. Examples of cubic modules

4.1. Pseudo-quadratic spaces

4.2. Adjoint action

4.3. The Tits-Kantor-Koecher module

4.4. Quadratic pairs without commuting root subgroups

4.5. Elementary groups of Freudenthal triple systems

4.6. Connection with Moufang Quadrangles

4.7. Suzuki and Ree groups

Chapter 5. The structure of a cubic module

Chapter 6. Construction of irreducible submodules

Chapter 7. Cubic rank one groups with trivial quadratic kernel

Chapter 8. A characterisation of the adjoint module of \PSL₂( )

Chapter 9. Cubic rank one groups with non-trivial quadratic kernel

Chapter 10. Cubic rank one groups with Hermitian quadratic kernel

Chapter 11. Cubic rank one groups with commutative quadratic kernel

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Grüninger, Matthias Cubic Action of a Rank One Group

ISBN:

9781470470227

OCLC:

1312158552