Hurwitz's theorem (composition algebras): Difference between revisions

 
===Definition===
A '''Hurwitz algebra''' or '''composition algebra''' is a finite-dimensional not necessarily associative algebra {{mvar|A}} with identity endowed with a nondegenerate quadratic form {{mvar|q}} such that {{math|1=''q''(''a b'') = ''q''(''a'') ''q''(''b'')}}. If the underlying coefficient field is the [[real number|reals]] and {{mvar|q}} is positive-definite, so that {{math|1=(''a'', ''b'') = {{sfrac|1|2}}[''q''(''a'' + ''b'') − ''q''(''a'') − ''q''(''b'')]}} is an [[inner product space|inner product]], then {{mvar|A}} is called a '''Euclidean Hurwitz algebra''' or (finite-dimensional) '''normed division algebra'''.<ref>{{harvnb|Faraut|Koranyi|1994|p=82}}</ref>
 
If {{mvar|A}} is a Euclidean Hurwitz algebra and {{mvar|a}} is in {{mvar|A}}, define the involution and right and left multiplication operators by