Quadrangular Algebras. (MN-46)

Quadrangular Algebras. (MN-46)

Richard M. Weiss
Copyright Date: 2006
Pages: 144
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    Quadrangular Algebras. (MN-46)
    Book Description:

    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 E8as 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 Algebrasis 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.

    eISBN: 978-1-4008-2694-0
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Preface
    (pp. vii-x)
  4. Chapter One Basic Definitions
    (pp. 1-10)

    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. Aquadratic spaceis a triple$(K,{\rm{ }}L,{\rm{ }}q)$, 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)$q\left( {u + v} \right) = q\left( u \right) + q\left( v \right) + f\left( {u,v} \right)$and

    (ii)$q\left( {tu} \right) - {t^2}q\left( u \right)$

    for all$u,v \in L$and all$t \in K,$, where$f$is a bilinear form on$L$(i.e. a symmetric bilinear map from$L \times L to K)$. A quadratic space$(K,{\rm{ }}L,{\rm{ }}q)$is calledanisotropicif

    $q\left( u \right) = 0{\text{ and only if }}u = 0$.


  5. Chapter Two Quadratic Forms
    (pp. 11-20)

    In this chapter we assemble the facts and definitions from the theory of quadratic forms which we will need in the study of quadrangular algebras.

    Definition 2.1. Anisomorphismfrom one quadratic space$(K,{\rm{ }}L,{\rm{ }}q)$to another$(\hat K,\;\hat L,\;\hat q)$is a pair of maps$({\psi _0},\;{\psi _1})$such that${\psi _0}$is an isomorphism from$K$to$\hat K,\;{\psi _1}$is a${\psi _0}$-linear isomorphism from$L$to${\hat L}$and$\hat q({\psi _1}(u))\; = \;{\psi _0}(q(u))$for all$u\; \in \;L$. Two quadratic spaces areisomorphicif there is an isomorphism from one to the other. We will only use the symbol$ \cong $when$K\; = \;\hat K$and there is an isomorphism$({\psi _0},\;{\psi _1})$such that${\psi _0}$...

  6. Chapter Three Quadrangular Algebras
    (pp. 21-28)

    In this chapter and the next, we assemble the basic facts about quadrangular algebras which we will need to carry out their classification. The classification of proper quadrangular algebras is comprised of three theorems which we state here so that the reader knows where we are headed in Chapters 5–7. (We recall that the termsproper,regular,defectiveandspecialare defined in 1.27 and 1.28. Note, too, that by 2.5, special implies regular. Quadrangular algebras which are not proper are classified in Chapter 9.)

    Theorem 3.1.Let

    $\Xi = (K,\;L,\;q,\;1,\;X,\; \cdot ,\;h,\;\theta )$

    be a proper quadrangular algebra such that

    ${\rm{di}}{{\rm{m}}_K}L\; \le \;4$.

    Then$\Xi $...

  7. Chapter Four Proper Quadrangular Algebras
    (pp. 29-36)

    We continue introducing the most fundamental properties of quadrangular algebras. In this chapter, we focus onproperquadrangular algebras (as defined in 1.27).

    Definition 4.1. Let$\Xi \; = \;(K,\; \ldots ,\;\theta )$be a quadrangular algebra, let$f$and$\pi $be as in 1.17 and let$\delta \; \in \;L$. Then$\Xi $is$\delta {\rm{ - standard}}$whenever the following hold:

    (i)$\delta = 1\,/\,2$(in$K\; \subset \;L$) if char${\rm{(}}K) \ne \;2$

    (ii)$f(1,\;\delta ) = 1$if char${\rm{(}}K)\; = \;2$and

    (iii)$f(\pi (a),\;\delta )\; = \;0$for all$a\; \in \;X$in all characteristics.

    We will say that$\Xi $isstandardif char$(K) \ne \:2$and$\Xi $is$\delta {\rm{ - }}standard$for$\delta = 1\,/\,2$(the onlly choice).

    Note that 4.1.i implies that$f(1,\;\delta ) = 1$also if char$(K) \ne \:2$. If char${\rm{(}}K)\; = \;2$, then...

  8. Chapter Five Special Quadrangular Algebras
    (pp. 37-44)

    We continue to assume that

    $\Xi \; = \;(K,\;L,\;q,\;1,\;X,\; \cdot ,\;h,\;\theta )$

    is a$\delta {\rm{ - }}$standard quadrangular algebra for some$\delta \; \in \;L$(as defined in 4.1) and we continue to let$f,\;\sigma ,\;g,\;\phi $and$\pi $be as in 1.17. In this chapter, we assume as well that

    (5.1)${\rm{di}}{{\rm{m}}_K}L\; \le \;4$.

    Our goal is to show that under these assumptions,$\Xi $is special (as defined in 1.28).

    Proposition 5.2.Let$a\; \in \;X{\kern 1pt} *$. Then either${\dim _K}L\: = \:2$and$L\; = \;\langle 1,\;\pi (a)\rangle $or${\dim _K}L\: = \:4.$Suppose that${\dim _K}L\: = \:4.$. Then

    (i)$L\; = \;\langle 1,\;\pi (a),\;w,\;\theta (a,\;w)\rangle$and$\langle w,\;\theta (a,\;w)\rangle \; = \;{\langle 1,\;\pi (a)\rangle ^ \bot }$for all non-zero elements$w\; \in \;{\langle 1,\;\pi (a)\rangle ^ \bot }$if either${\text{char}}(K) \ne 2$or${\text{char}}(K)\: = \:2$and$f(\pi (a),\;1)\; \ne \;0$.

    (ii)$L\; = \;\langle 1,\;\delta ,\;\pi (a),\;\theta (a,\;\delta )\rangle$and

    $\langle \pi (a),\;\theta (a,\;\delta )\rangle \; = \;{\langle 1,\;\delta \rangle ^ \bot }$

    if${\text{char}}(K)\: = \:2$and$f(\pi (a),\;1)\; = \;0$.

    Proof. By D2,$\langle 1,\;\pi (a)\rangle$is...

  9. Chapter Six Regular Quadrangular Algebras
    (pp. 45-58)

    We continue to assume that

    $\Xi \; = \;(K,\;L,\;q,\;1,\;X,\; \cdot ,\;h,\;\theta )$

    is a$\delta {\rm{ - }}$standard quadrangular algebra (for some$\delta \; \in \;L$) and we continue to let$f,\;\sigma ,\;g,\;\phi$and$\pi$be as in 1.17. In this chapter, we assume as well that$\Xi$is regular (as defined in 1.27) but not special (as defined in 1.28). Our goal is to show that$\left( {K,L,q} \right)$must be a quadratic space of type${E_6},\;{E_7}\;{\rm{or}}\;{E_8}$(as defined in 2.13) and that$\Xi$is uniquely determined by the pointed quadratic space$(K,L,q,1)$.

    Since$\Xi$is not special, we have

    (6.1)${\dim _K}L > {\text{4}}$

    by 5.9.

    Proposition 6.2.If${\text{char}}(K)\: = \:2$then there exists an element...

  10. Chapter Seven Defective Quadrangular Algebras
    (pp. 59-76)

    We continue to assume that

    $\Xi \; = \;(K,\;L,\;q,\;1,\;X,\; \cdot ,\;h,\;\theta )$

    is a$\delta {\rm{ - }}$standard quadrangular algebra (for some$\delta \; \in \;L$) and we continue to let$f,\;\sigma ,\;g,\;\phi$and$\pi$be as in 1.17. In this chapter, we assume in addition that the defect${L^ \bot }$of$\left( {K,L,q} \right)$(as defined in 2.3) is non-trivial. Our goal is to show that$\left( {K,L,q} \right)$must be a quadratic space of type${F_4}$(as defined in 2.15) and that$\Xi$is uniquely determined by the pointed quadratic space$\left( {K,L,q,1} \right)$.

    Let$R = {L^ \bot }.{\text{e}}$By 2.4, we have${\text{char}}(K)\: = \:2.$By 4.1.ii, we have$f(1,\;\delta )\; = \;1.$By 1.2, therefore,${\delta ^\sigma } = \delta \; + \;1$. Thus, in particular,$\sigma \ne 1.$This means that$\Xi$is...

  11. Chapter Eight Isotopes
    (pp. 77-82)

    In the geometrical considerations which gave rise to the notion of a quadrangular algebra, there is nothing canonical about the basepoint. To capture this idea algebraically, we introduce in this chapter the notion ofisotopicquadrangular algebras.

    Recall that by 3.14 and 4.2, it is no real restriction to assume that a quadrangular algebra is standard (as defined in 4.1) if the characteristic is different from two.

    Proposition 8.1.Let

    $\Xi \; = \;(K,\;L,\;q,\;1,\;X,\; \cdot ,\;h,\;\theta )$

    be a quadrangular algebra and let$\sigma , g$and$\phi$be as in 1.17. Suppose that$\Xi$is standard if${\text{char}}(K) \ne 2$(but if${\text{char}}(K)\: = \:2,$$\Xi$may or may not be...

  12. Chapter Nine Improper Quadrangular Algebras
    (pp. 83-94)

    In this chapter we show that improper quadrangular algebras are classified by certain indifferent sets (defined in 9.28 below) and certain anisotropic quadratic spaces. First, we have to be a little careful with the definition of “improper.”

    Proposition 9.1.Let

    $\Xi \; = \;(K,\;L,\; \ldots ,\;\theta )$

    be a quadrangular algebra and let f be as in 1.17. Then f is not identically zero if and only if some isotope of$\Xi$is proper.

    Proof. If${\text{char}}(K) \ne 2$then$f(u,\;u)\; = \;2q(u) \ne \;0$for all$u\; \in \;L\,*$and$\Xi$is proper by 3.14. We can assume, therefore, that${\text{char}}(K)\: = \:2.$Suppose that$f(u,\;L) \ne 0$for some$u\; \in \;L.$Replacing$\Xi$by${\Xi _u}$as defined...

  13. Chapter Ten Existence
    (pp. 95-108)

    In this chapter, we give a proof of the following result:

    Theorem 10.1.Let$\left( {K,L,q} \right)$be a quadratic space of type${E_6},\;{E_7},\;{E_8}\;or\;{F_4}$and let$u\; \in \;L\,*.$Then there exist$X,.,h$and$\theta$such that

    $\Xi \; = \;(K,\;L,\;q\,/\,q(u),\;u,\;X,\; \cdot ,\;h,\;\theta )$

    is a quadrangular algebra.

    By 8.1, it suffices to show in 10.1 that$\Xi$exists forsome$u\; \in \;L\,*.$

    We have already shown in 3.2 and 3.3 that the quadrangular algebra$\Xi$in 10.1 is unique.

    The proof of 10.1 we give is based on results in Chapters 13 and 14 of [12]. Our strategy is to construct a right$C(q\,/\,q(u),\;u){\rm{ - module}}$$X$explicitly (using coordinates), give formulas...

  14. Chapter Eleven Moufang Quadrangles
    (pp. 109-124)

    In this chapter, we describe the connection between quadrangular algebras and Moufang quadrangles. From 11.15 on, we examine the connection between quadrangular algebras and the stabilizer of an apartment in the automorphism group of an exceptional Moufang quadrangle (and continue with this theme in the next chapter).

    Let$\Gamma $be a generalized$n$-gon (for some$n\; \ge \;3$).¹ Thus$\Gamma $is a bipartite graph of diameter$n$whose minimal circuits have length$2n$. We will always assume that$\Gamma $isthick. This means that$\,{\Gamma _x}\,|\; \ge \;3$for all vertices$x$, where${\Gamma _x}$denotes the set of vertices adjacent to$x$.

    A path of...

  15. Chapter Twelve The Structure Group
    (pp. 125-132)

    The structure group of a quadrangular algebra is essentially the automorphism group of its isotopy class. In 12.11 below, we show that the structure group of an exceptional quadrangular algebra is isomorphic to the group of “linear” elements in the pointwise stabilizer of an apartment in the automorphism group of the corresponding Moufang quadrangle. (This is the group we called${H_0}$in Chapter 11; see 11.20.)


    $\Xi \; = \;(K,\;L,\;q,\;1,\;X,\; \cdot ,\;h,\;\theta )$

    be a quadrangular algebra which we assume to be standard (as defined in 4.1) if${\text{char}}(K) \ne \:2.$Let [$\Xi$] denote the set

    $\{ {\Xi _u}|u\; \in \;L*\} $,

    where the${\Xi _u}$are the isotopes of$\Xi$as defined...

  16. Bibliography
    (pp. 133-134)
  17. Index
    (pp. 135-135)