Quaternionic structure

In mathematics, a quaternionic structure or Q-structure is an axiomatic system that abstracts the concept of a <a href="/facts/Quaternion_algebra/CYtNzofE">quaternion algebra</a> over a <a href="/facts/Field_(algebra)/xAjAS4ko">field</a>.
A quaternionic structure is a triple (G, Q, q) where G is an <a href="/facts/Elementary_abelian_group/rWVah6P0">elementary abelian group</a> of <a href="/facts/Exponent_(group_theory)/TEvLhDtm">exponent</a> 2 with a distinguished element −1, Q is a <a href="/facts/Pointed_set/EELNN0pq">pointed set</a> with distinguished element 1, and q is a symmetric <a href="/facts/Surjection/KezhLpCb">surjection</a> G×G → Q satisfying axioms

1.
                
                
              
              
                q
                (
                a
                ,
                (
                −
                1
                )
                a
                )
                =
                1
                ,
              
            
            
              
                
                  2.
                
                
              
              
                q
                (
                a
                ,
                b
                )
                =
                q
                (
                a
                ,
                c
                )
                ⇔
                q
                (
                a
                ,
                b
                c
                )
                =
                1
                ,
              
            
            
              
                
                  3.
                
                
              
              
                q
                (
                a
                ,
                b
                )
                =
                q
                (
                c
                ,
                d
                )
                ⇒
                ∃
                x
                ∣
                q
                (
                a
                ,
                b
                )
                =
                q
                (
                a
                ,
                x
                )
                ,
                q
                (
                c
                ,
                d
                )
                =
                q
                (
                c
                ,
                x
                )
              
            
          
        
        .
      
    
    {\displaystyle {\begin{aligned}{\text{1.}}\quad &q(a,(-1)a)=1,\\{\text{2.}}\quad &q(a,b)=q(a,c)\Leftrightarrow q(a,bc)=1,\\{\text{3.}}\quad &q(a,b)=q(c,d)\Rightarrow \exists x\mid q(a,b)=q(a,x),q(c,d)=q(c,x)\end{aligned}}.}

Every field F gives rise to a Q-structure by taking G to be F∗/F∗2, Q the set of <a href="/facts/Brauer_class/dQQbnlsp">Brauer classes</a> of quaternion algebras in the <a href="/facts/Brauer_group/dQQbnlsp">Brauer group</a> of F with the <a href="/facts/Split_quaternion_algebra/CYtNzofE">split quaternion algebra</a> as distinguished element and q(a,b) the quaternion algebra (a,b)F.

<ul><li><a href="/facts/T._Y._Lam/S1ewInkE">Lam, Tsit-Yuen</a> (2005). Introduction to Quadratic Forms over Fields. <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>. Vol. 67. <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-1095-2. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2104929">2104929</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1068.11023">1068.11023</a>.</li></ul>

Quaternionic structure open-in-new

Quaternionic structure