Norm (mathematics): Difference between revisions

The concept of norm <math>N(z)</math> in [[composition algebra]]s does {{em|not}} share the usual properties of a norm assince it[[null mayvector]]s beare negative or zero for <math>z \neq 0allowed.</math> A composition algebra <math>(A, {}^*, N)</math> consists of an [[algebra over a field]] <math>A,</math> an [[involution (mathematics)|involution]] <math>{}^*,</math> and a [[quadratic form]] [[Degree of a field extension|<math>N(z) = z z^*</math>]] called the "norm".

The characteristic feature of composition algebras is the [[homomorphism]] property of <math>N</math>: for the product <math>w z</math> of two elements <math>w</math> and <math>z</math> of the composition algebra, its norm satisfies <math>N(wz) = N(w) N(z).</math> ForIn the case of [[division algebra]]s <math>\R,</math> <math>\Complex,</math> <math>\mathbb{H},</math> and '''O''' the composition algebra norm is the square of the norm discussed above. In those cases the norm is a [[definite quadratic form]]. In otherthe composition[[split algebrasalgebra]]s the norm is an [[isotropic quadratic form]].