Jacobson’s theorem on composition algebras
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 .
Theorem 1 (Jacobson).
[1, Theorem 3.23]
Two unital Cayley-Dickson algebras and over a field of characteristic not
are isomorphic if, and only if, their quadratic forms are isometric.
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 .
References
- 1 Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York, 1966.
Title | Jacobson’s theorem on composition algebras |
---|---|
Canonical name | JacobsonsTheoremOnCompositionAlgebras |
Date of creation | 2013-03-22 17:18:14 |
Last modified on | 2013-03-22 17:18:14 |
Owner | Algeboy (12884) |
Last modified by | Algeboy (12884) |
Numerical id | 4 |
Author | Algeboy (12884) |
Entry type | Theorem |
Classification | msc 17A75 |
Related topic | CompositionAlgebrasOverMathbbR |
Related topic | HurwitzsTheoremOnCompositionAlgebras |
Related topic | CompositionAlgebraOverAlgebaicallyClosedFields |
Related topic | CompositionAlgebrasOverFiniteFields |
Related topic | CompositionAlgebrasOverMathbbQ |