Quadrangular Algebras. (MN-46) (Mathematical Notes, 64, Band 64) - Softcover

Weiss, Richard M.

Softcover

ISBN 10: 0691124604 ISBN 13: 9780691124605

Verlag: Princeton University Press, 2005

Inhaltsangabe

This book introduces a new class of non-associative algebras related to certain exceptional algebraic groups and their associated buildings. Richard Weiss develops a theory of these "quadrangular algebras" that opens the first purely algebraic approach to the exceptional Moufang quadrangles. These quadrangles include both those that arise as the spherical buildings associated to groups of type E6, E7, and E8 as well as the exotic quadrangles "of type F4" discovered earlier by Weiss. Based on their relationship to exceptional algebraic groups, quadrangular algebras belong in a series together with alternative and Jordan division algebras. Formally, the notion of a quadrangular algebra is derived from the notion of a pseudo-quadratic space (introduced by Jacques Tits in the study of classical groups) over a quaternion division ring. This book contains the complete classification of quadrangular algebras starting from first principles. It also shows how this classification can be made to yield the classification of exceptional Moufang quadrangles as a consequence. The book closes with a chapter on isotopes and the structure group of a quadrangular algebra. Quadrangular Algebras is intended for graduate students of mathematics as well as specialists in buildings, exceptional algebraic groups, and related algebraic structures including Jordan algebras and the algebraic theory of quadratic forms.

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Über die Autorin bzw. den Autor

Richard M. Weiss

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Quadrangular Algebras

By Richard M. Weiss

PRINCETON UNIVERSITY PRESS

Copyright © 2006 Princeton University Press
All right reserved.
ISBN: 978-0-691-12460-5

Contents

Preface.........................................................viiChapter 1. Basic Definitions...................................1Chapter 2. Quadratic Forms.....................................11Chapter 3. Quadrangular Algebras...............................21Chapter 4. Proper Quadrangular Algebras........................29Chapter 5. Special Quadrangular Algebras.......................37Chapter 6. Regular Quadrangular Algebras.......................45Chapter 7. Defective Quadrangular Algebras.....................59Chapter 8. Isotopes............................................77Chapter 9. Improper Quadrangular Algebras......................83Chapter 10. Existence...........................................95Chapter 11. Moufang Quadrangles.................................109Chapter 12. The Structure Group.................................125Bibliography....................................................133Index...........................................................134

Chapter One

Basic Definitions

Before we can give the definition of a quadrangular algebra (in 1.17 below), we need to review a few standard notions.

Definition 1.1. A quadratic space is a triple (K, L, q), where K is a (commutative) field, L is a vector space over K and q is a quadratic form on L, that is, a map from L to K such that

(i) q(u + v) = q(u) + q(v) + f(u, v) and

(ii) q(tu) = [t.sup.2]q(u)

for all u, v [member of] L and all t [member of] K, where f is a bilinear form on L (i.e. a symmetric bilinear map from L x L to K). A quadratic space (K, L, q) is called anisotropic if

q(u) = 0 if and only if u = 0.

A basepoint of a quadratic space (K, L, q) is an element 1 of L such that

q(1) = 1.

A pointed quadratic space is a quadratic space with a distinguished basepoint. Suppose that (K, L, q, 1) is a pointed quadratic space (with basepoint 1). The standard involution of (K, L, q, 1) is the map [sigma] from L to itself given by

(1.2) [u.sup.[sigma]] = f(1, u)1 - u

for all u [member of] L, where f is as in (i) above. We set

(1.3) [v.sup.-1] = [v.sup.[sigma]]/q(v)

for all v such that q(v) [not equal to] 0. We will always identify K with its image under the map t [??] t 1 from K to L.

Note that if q, f and [sigma] are as in 1.1 then [[sigma].sup.2] = 1 and q([u.sup.[sigma]]) = q(u) for all u [member of] L. It follows that

(1.4) f([u.sup.[sigma]], [v.sup.[sigma]]) = f(u, v)

for all u, v [member of] L as well as

(1.5) q([v.sup.-1]) = q[(v).sup.-1]

and

(1.6) [([v.sup.-1]).sup.-1] = v

for all v [member of] [L.sup.*].

Definition 1.7. Let L be an arbitrary ring. An involution of L is an anti-automorphism [sigma] of L such that [[sigma].sup.2] = 1. For each involution [sigma] of L, we set

[L.sub.[sigma]] = {u + [u.sup.[sigma]] | u [member of] L}.

The elements of [L.sub.[sigma]] are called traces (with respect to [sigma]).

Note that in 1.2 we have called [sigma] the standard involution of the pointed quadratic space (K,L, q, 1) even though it is not, according to 1.7, really an involution (since there is no multiplication on L). The next two definitions will make it clear why we have done this.

Definition 1.8. Let E/K be a separable quadratic field extension, let N denote its norm and let [sigma] denote the non-trivial element of Gal(E/K), so N(u) = [uu.sup.[sigma]] for all u [member of] E. Let [alpha] [member of] [K.sup.*] and let M(2,E) denote the K-algebra of 2 x 2 matrices over E. The quaternion algebra (E/K,[alpha]) is the subalgebra

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of M(2,E). Let L = (E/K,[alpha]).

We identify E (and thus also K [subset] E) with its image in L under the map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then every element of L can be written uniquely in the form u+ev with u, v [member of] E and multiplication in L is determined by associativity, the distributive laws and the identities [e.sup.2] = [alpha] and ue = [eu.sup.[sigma]] for all u [member of] E. The extension of [sigma] [member of] Gal(E/K) to the map from L to itself (which we also denote by [sigma]) given by

(1.9) [(u + ev).sup.[sigma]] = [u.sup.[sigma]] - ev

for all u, v [member of] E is an involution of L (as defined in 1.7) called the standard involution of L. Both the center Z(L) of L and the set of traces [L.sub.[sigma]] (as defined in 1.7) equal K and [ww.sup.[sigma]] [member of] K for all w [member of] L. The extension of the norm N to a map from L to K (which we also denote by N) given by

N(w) = [ww.sup.[sigma]]

for all w [member of] L is a quadratic form on L (as a vector space over K) called the reduced norm of L. An element w of L is invertible if and only if N(w) [not equal to] 0, in which case

(1.10) [w.sup.-1] = [w.sup.[sigma]]/N(w).

In particular, L is a division algebra if and only if the reduced norm N is anisotropic (as defined in 1.1). Since

(1.11) N(u + ev) = N(u) - [alpha]N(v)

for all u, v [member of] E, it follows that L is a division algebra (i.e. a skew field) if and only if [alpha] [not member of] N (E).

Definition 1.12. Let L be a skew field, let [sigma] be an involution of L and let K = [L.sub.[sigma]] (as defined in 1.7). We will say that the pair (L, [sigma]) is quadratic if either L is commutative and [sigma] [not equal to] 1 (in which case K is a subfield of L, L/K is a separable quadratic extension and [sigma] is the unique non-trivial element in Gal[(L/K).sup.1]) or L is quaternion and [sigma] is its standard involution as defined in 1.9 (in which case K = Z(L)). Suppose that (L, [sigma]) is quadratic and let

(1.13) q(u) = [uu.sup.[sigma]]

for all u [member of] L (so q is the norm of the extension L/K if L is commutative and q is the reduced norm of L if L is quaternion). Then (K,L, q, 1) is a pointed anisotropic quadratic space and

(1.14) f(u, v) = [uv.sup.[sigma]] + [vu.sup.[sigma]]

for all u, v [member of] L, where f is as in 1.1.i. By 1.2, it follows that [sigma] is the standard involution of (K,L, q, 1) and by 1.10, the element [u.sup.-1] defined in 1.3 is, in fact, the inverse of u in the skew field L for all u [member of] [L.sup.*].

The following classical result is attributed...

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2005
• Sprache: Englisch
• ISBN 10: 0691124604
• ISBN 13: 9780691124605
• Einband: Taschenbuch
• Anzahl der Seiten: 146
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: gpsr@libri.de; gpsr@libri.de; ; Friedensallee 273; Hamburg; 22763; Deutschland