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

Weiss, Richard M.

Softcover

ISBN 10: 0691124604 ISBN 13: 9780691124605

Publisher: Princeton University Press, 2005

Synopsis

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.

About this title

About the Author

Richard M. Weiss is William Walker Professor of Mathematics at Tufts University. He is the author of "The Structure of Spherical Buildings" (Princeton) and the coauthor (with Jacques Tits) of "Moufang Polygons". He received a Humboldt Research Prize in 2004.

Excerpt. © Reprinted by permission. All rights reserved.

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 (but only in characteristic different from two) in (page 187) to J. Dieudonn. The general case can be found in Theorem 2.1.10 of. We mention this result only to indicate that it is well known that among skew fields with involution, those singled out in 1.12 are exceptional.

Theorem 1.15. Let L be a skew field with a non-trivial involution [sigma] and let [L.sub.[sigma]] be as in 1.7. Then either L is generated by [L.sub.[sigma]] as a ring or the pair (L, [sigma]) is quadratic as defined in 1.12.

Definition 1.16. A (skew-hermitian) pseudo-quadratic space is a set

(L, [sigma],X, h, [pi]),

where L is a skew field, [sigma] is an involution of L (as defined in 1.7), X is a right vector space over L, h is a skew-hermitian form on X (i.e. h is a bi-additive map from X x X to L such that

(i) h(a, bu) = h(a, b)u and

(ii) h[(a, b).sup.[sigma]] = -h(b, a)

for all a, b [member of] X and all u [member of] L) and [pi] is a map from X to L such that

(iii) [pi](a + b) [equivalent to] [pi](a) + [pi](b) + h(a, b) (mod [L.sub.[sigma]]) and

(iv) [pi](au) [equivalent to] [u.sup.[sigma]][pi](a)u (mod [L.sub.[sigma]])

for all a, b [member of] X and all u [member of] L, where [L.sub.[sigma]] is as in 1.7. A pseudo-quadratic space

(L, [sigma],X, h, [pi])

is called anisotropic if

(v) [pi](a) [equivalent to] 0 (mod [L.sub.[sigma]]) only if a = 0.

and standard if

(iv') [pi](au) = [u.sup.[sigma]][pi](a)u

for all a [member of] X and all u [member of] L.

Let

[PI] = (L, [sigma],X, h, [pi])

be an arbitrary pseudo-quadratic space. By (11.28) and (11.31) of, there exists a map [??] from X to L such that

(i) [??](a) [equivalent to] [pi](a) (mod [L.sub.[sigma]]) for all a [member of] X (so

[??] = (L, [sigma],X, h, [??])

is also a pseudo-quadratic space which is, in an obvious sense, equivalent to [PI]) and

(ii) [??] is standard (as defined in 1.16.iv').

Here now is the main definition of this monograph.

Definition 1.17. A quadrangular algebra is a set

(K,L, q, 1,X, , h, [theta]),

where (K,L, q, 1) is a pointed anisotropic quadratic space as defined in 1.1, X is a non-trivial vector space over K, (a, v) [??] a v is a map from X x L to X (which is denoted below, and, in general, simply by juxtaposition), h is a map from X x X to L and [theta] a map from X x L to L satisfying the following twelve axioms (in which f is as in 1.1.i, [sigma] is as in 1.2 and [v.sup.-1] is as in 1.3):

(A1) The map is bilinear (over K).

(A2) a 1 = a for all a [member of] X.

(A3) (av)[v.sup.-1] = a for all a [member of] X and all v [member of] [L.sup.*].

(B1) h is bilinear (over K).

(B2) h(a, bv) = h(b, av) + f(h(a, b), 1) v for all a, b [member of] X and all v [member of] L.

(B3) f(h(av, b), 1) = f(h(a, b), v) for all a, b [member of] X and all v [member of] L.

(C1) For each a [member of] X, the map v [??] [theta](a, v) is linear (over K).

(C2) [theta](ta, v) = [t.sup.2][theta](a, v) for all t [member of] K, all a [member of] X and all v [member of] L.

(C3) There exists a function g from X x X to K such that

[theta](a + b, v) = [theta](a, v) + [theta](b, v) + h(a, bv) - g(a, b)v

for all a, b [member of] X and all v [member of] L.

(C4) There exists a function [phi] from X x L to K such that

[theta](av,w) = [theta][(a,[w.sup.[sigma]]).sup.[sigma]] q(v) - f(w, [v.sup.[sigma]])[theta][(a, v).sup.[sigma]] + f([theta](a, v), [w.sup.[sigma]])[v.sup.[sigma]] + [phi](a, v)w

for all a [member of] X and v,w [member of] L.

(D1) Let [pi](a) = [theta](a, 1) for all a [member of] X. Then

a[theta](a, v) = (a[pi](a))v

for all a [member of] X and all v [member of] L and

(D2) [pi](a) [equivalent to] 0 (mod K) if and only if a = 0 (where K has been identified with its image under the map t [??] t 1 from K to L).

As was indicated in the introduction, the definition of a quadrangular algebra is derived from the definition of an anisotropic pseudo-quadratic space defined over a skew field with involution (L, [sigma]) which is quadratic as defined in 1.12. Roughly speaking, the axioms A1-D2 capture the structure which remains when the multiplication on L is discarded. We make these comments more precise with the following result:

Proposition 1.18. Let

[PI] = (L, [sigma],X, h, [pi])

be a standard anisotropic pseudo-quadratic space as defined in 1.16, suppose that the pair (L, [sigma]) is quadratic as defined in 1.12 and let K = [L.sub.[sigma]]. Let denote the scalar multiplication from X x L to X (so X is a vector space over K with respect to the restriction of to X xK), let q be as in 1.12 (i.e. q is the norm of L/K if L is commutative and q is the reduced norm of L if L is quaternion) and let

[theta](a, u) = [pi](a)u

for all a [member of] X and all u [member of] L. Then

(K,L, q, 1, X, , h, [theta])

is a quadrangular algebra with [phi] identically zero (where [phi] is as in axiom C4).

Proof. As observed in 1.12, (K,L, q, 1) is an anisotropic pointed quadratic space and [sigma] is its standard involution (as defined in 1.1). Axioms A1-A3 hold since X is a right vector space over L, K [subset] Z(L) and [v.sup.-1] = [v.sup.[sigma]]/q(v) in the inverse of v in the skew field L for all v [member of] [L.sup.*].

Let f denote the bilinear form associated with q. Since q([u.sup.[sigma]]) = q(u) for all u [member of] L, we have q(u) = u[u.sup.[sigma]] = [u.sup.[sigma]]u and thus (by 1.1.i)

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

for all u, v [member of] L. By 1.16.i and 1.16.ii (and 1.7), we have

(1.20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all a, b [member of] X and all v [member of] L. Thus B1 holds by 1.16.i, 1.20 (since [sigma] acts trivially on K [subset] L) and the assumption in 1.16 that h is bi-additive. Moreover,

h(a, bv) - h(b, av) = (h(a, b) - h(b, a))v by 1.16.i = (h(a, b) + h[(a, b).sup.[sigma]])v by 1.16.ii = f(h(a, b), 1)v by 1.19

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all a, b [member of] X and all v [member of] L. Thus axioms B2 and B3 hold. Axioms C1 and D1 are obvious. Axiom C2 holds by 1.16.iv' (and the assumption that ? is standard) since K [subset] Z(L) and [sigma] acts trivially on K. Axiom C3 holds by 1.16.iii and axiom D2 holds by 1.16.v. It remains only to prove that C4 holds. Choose a [member of] X and v, w [member of] L. We first observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by 1.7 and 1.19. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since

[w.sup.[sigma]][[v.sup.[sigma]] + vw [member of] [L.sub.[sigma]] [subset] Z(L),

it follows (by 1.19) that

(1.21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus C4 holds with [phi] identically zero.

In Chapter 5 we will characterize the quadrangular algebras which arise from anisotropic pseudo-quadratic spaces as in 1.18; see 1.28 and 3.1.

Here are some further comments about the definition 1.17 of a quadrangular algebra:

(i) We will, in general, refer to the axioms in 1.17 by their names-A1, A2, etc.-without referring explicitly to 1.17.

(ii) We have put the axioms of a quadrangular algebra into four groups in the hope of making them easier to recall. Axioms A1-A3 establish properties of the "scalar multiplication" (a, v) [??] av (those properties which remain after the multiplication on L is taken away). Axioms B1-B3 establish properties of the "skew-hermitian form" h. Axioms C1-C4 and D1-D2 establish properties of the "quasi-pseudo-quadratic form" [pi] and its companion [theta]. Axioms D1 and D2 are somehow different from the others and have been listed separately. The "polarization" of axiom D1 given in 3.22 below will play an especially important role in the classification of quadrangular algebras.

(iii) Among the axioms of a quadrangular algebra, C4 is conspicuously less "natural"-looking than the others. In the proof of 1.18, however, we saw that C4 is just 1.16.iv' re-written in a way that avoids the multiplication on L.

(iv) We emphasize that there is no multiplication on L in 1.17. Thus if a [member of] X and u, v [member of] L, then auv can only mean (au)v, and we will almost always, in fact, omit the parentheses in such an expression. Similarly, auvw can only mean ((au)v)w, etc.

(Continues...)

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Edition details

• Publisher: Princeton University Press
• Publication date: 2005
• Language: English
• ISBN 10: 0691124604
• ISBN 13: 9780691124605
• Binding: Paperback
• Number of pages: 146