Composition algebra

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a composition algebra A over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K is a <a href="/facts/Non-associative_algebra/Vk9Kjmrs">not necessarily associative</a> <a href="/facts/Algebra_over_a_field/8T8ulWJb">algebra</a> over K together with a <a href="/facts/Degenerate_form/9TLGBvSH">nondegenerate</a> <a href="/facts/Quadratic_form/l2n1jXPe">quadratic form</a> N that satisfies

N
        (
        x
        y
        )
        =
        N
        (
        x
        )
        N
        (
        y
        )
      
    
    {\displaystyle N(xy)=N(x)N(y)}

for all x and y in A.
A composition algebra includes an <a href="/facts/Involution_(mathematics)/kKxYrUVa">involution</a> called a conjugation: 
 
 
 
 x
 ↦
 
 x
 
 ∗
 
 
 .
 
 
 {\displaystyle x\mapsto x^{*}.}
 
 The quadratic form 
 
 
 
 N
 (
 x
 )
 =
 x
 
 x
 
 ∗
 
 
 
 
 {\displaystyle N(x)=xx^{*}}
 
 is called the norm of the algebra.
A composition algebra (A, ∗, N) is either a <a href="/facts/Division_algebra/xp5Bepx3">division algebra</a> or a split algebra, depending on the existence of a non-zero v in A such that N(v) = 0, called a <a href="/facts/Null_vector/sjkdQtdS">null vector</a>. When x is not a null vector, the <a href="/facts/Multiplicative_inverse/yczdYyCw">multiplicative inverse</a> of x is 
 
 
 
 
 
 
 x
 
 ∗
 
 
 
 N
 (
 x
 )
 
 
 
 
 
 {\textstyle {\frac {x^{*}}{N(x)}}}
 
. When there is a non-zero null vector, N is an <a href="/facts/Isotropic_quadratic_form/vMMamxqE">isotropic quadratic form</a>, and "the algebra splits".

Composition algebra open-in-new

Composition algebra