Hurwitz algebra

English

Etymology

Named after German mathematician Adolf Hurwitz (1859–1919); see also Hurwitz problem, Hurwitz's theorem.

Noun

Hurwitz algebra (plural Hurwitz algebras)

1. (algebra) Any one of the unital composition algebras identified by Hurwitz's theorem (on composition algebras) as solutions to the Hurwitz problem.
   • 1995, Susumo Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge University Press, page 40:

     As we recall, a Hurwitz algebra is a composition algebra with unit element.

   • 1998, Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol, The Book of Involutions, American Mathematical Society, page 465:

     Observe that the unital composition algebra associated with $(C,\star )$ by the construction given in the proof of Proposition (33.27) is the Hurwitz algebra $(C,\diamond )$ if we set $a=1$.

   • 2006, Alberto Elduque, Chapter 12: A new look at Freudenthal's Magic Square, Lev Sabinin, Larissa Sbitneva, Ivan Shestakov (editors, Non-Associative Algebra and Its Applications, Taylor & Francis Group (Chapman & Hall/CRC), page 15,

     Unital composition algebras (or Hurwitz algebras) form a well-known class of algebras. Any Hurwitz algebra has a finite dimension equal to either 1, 2, 4, or 8. The two-dimensional Hurwitz algebras are the quadratic étale algebras over the ground field F, the four dimensional ones are the generalized quaternion algebras, and the eight dimensional Hurwitz algebras are called Cayley algebras, and are analog to the classical algebra of octonions (for a good survey of the latter, see [Bae02]).

Hyponyms

• Euclidean Hurwitz algebra

Derived terms

• para-Hurwitz algebra

Translations

unital composition algebra that is a solution to the Hurwitz problem

See also

• composition algebra

Further reading

• Hurwitz's theorem (composition algebras) on Wikipedia.Wikipedia
• Definite quadratic form on Wikipedia.Wikipedia
• Clifford algebra on Wikipedia.Wikipedia