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Jacobson's theorem on composition algebras
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(Theorem)
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Recall that composition algebra  over a field  is specified with a quadratic form  . Furthermore, two quadratic forms  and  are isometric if there exists an invertible linear map  such that
 for all  .
A Cayley-Dickson algebra is split if the algebra has non-trivial zero-divisors.
Corollary 2 [1, Corollary 3.24] Upto isomorphism there is only one split Cayley-Dickson algebra and the quadratic form has Witt index 4.
Over the real numbers instead of Witt index, we say the signature of the quadratic form is  .
This result is often used together with a theorem of Hurwitz which limits the dimensions of composition algebras to dimensions 1,2, 4 or 8. Thus to classify the composition algebras over a given field  of characteristic not 2, it suffices to classify the non-degenerate quadratic forms
 with  or  .
- 1
- Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York, 1966.
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"Jacobson's theorem on composition algebras" is owned by Algeboy.
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(view preamble)
Cross-references: non-degenerate quadratic forms, dimensions, limits, signature, real numbers, index, isomorphism, algebra, isomorphic, characteristic, Cayley-Dickson algebras, unital, invertible linear map, isometric, quadratic form, field, composition algebra
This is version 1 of Jacobson's theorem on composition algebras, born on 2007-06-23.
Object id is 9651, canonical name is JacobsonsTheoremOnCompositionAlgebras.
Accessed 407 times total.
Classification:
| AMS MSC: | 17A75 (Nonassociative rings and algebras :: General nonassociative rings :: Composition algebras) |
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Pending Errata and Addenda
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