Jacobson’s theorem on composition algebras
Recall that composition algebra^{} $C$ over a field $k$ is specified with a quadratic form^{} $q:C\to k$. Furthermore, two quadratic forms $q:C\to k$ and $r:D\to k$ are isometric if there exists an invertible linear map^{} $f:C\to D$ such that $r(f(x))=q(x)$ for all $x\in C$.
Theorem 1 (Jacobson).
[1, Theorem 3.23] Two unital Cayley-Dickson algebras $C$ and $D$ over a field $k$ of characteristic not $\mathrm{2}$ 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 $(4,4)$.
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 $k$ of characteristic not 2, it suffices to classify the non-degenerate quadratic forms $q:{k}^{n}\to k$ with $n=1,2,4$ or $8$.
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 |