Composition algebra
In mathematics, a composition algebra A over a field K is a unital (but not necessarily associative) algebra over K together with a nondegenerate quadratic form N which satisfies
for all x and y in A. The quadratic form N is often referred to as a norm on A. Composition algebras are also called normed algebras (not to be confused with normed algebras in the sense of functional analysis).
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[edit] Structure theorem
Every composition algebra over a field K can be obtained by repeated application of the Cayley–Dickson construction starting from K (if the characteristic of K is different from 2) or a 2-dimensional composition subalgebra (if char(K) = 2). The possible dimensions of a composition algebra are 1, 2, 4, and 8.[1]
- 1-dimensional composition algebras only exist when char(K) ≠ 2.
- Composition algebras of dimension 1 and 2 are commutative and associative.
- Composition algebras of dimension 2 are either quadratic field extensions of K or isomorphic to
. - Composition algebras of dimension 4 are called quaternion algebras. They are associative but not commutative.
- Composition algebras of dimension 8 are called octonion algebras. They are neither associative nor commutative.
.[edit] Instances and usage
When the field K is taken to be complex numbers C, then the four composition algebras over C are C itself, the direct sum 
known first as tessarines (1848), the 2 x 2 complex matrix ring M(2,C), and the complex octonions CO.
Matrix ring M(2,C) has long been an object of interest, first as biquaternions by Hamilton (1853), later in the isomorphic matrix form, and especially as Pauli algebra. Complex octonions have been used in a model of angular momentum.[2]
The squaring function N(x) = x2 on the real number field forms the primordial composition algebra. When the field K is taken to be real numbers R, then there are just six other real composition algebras.[3]
In two, four, and eight dimensions there are both a "split algebra" and a "division algebra": complex numbers and split-complex numbers, quaternions and split-quaternions, octonions and split-octonions. These algebras find use in physical models. For instance, potential theory in the plane is based on the Laplace equation, a property of components of differentiable functions on complex numbers. The split-complex number multiplication is used for spacetime transformation. The indefinite orthogonal groups SO(2,2) of split-quaternions and SO(4,4) of split-octonions have been used for a quark model.[4]
[edit] See also
[edit] References
- ^ Jacobson 1958, Roos 2008
- ^ J. Koeplinger & V. Dzhunushaliev (2008) "Nonassociative decomposition of angular momentum operator using complex octonions", presentation at a meeting of the American Physical Society
- ^ Guy Roos (2008) Theorem 1.10 page 166
- ^ P. Alberca Bjerregard & C. Martin Gonzalez (1996) "Automorphism groups of composition algebras and quark models", Hadronic Journal 19:637–53
- Adrian Albert (1942) "Quadratic forms permitting composition", Annals of Mathematics 43:161–77.
- Nathan Jacobson (1958) "Composition algebras and their automorphisms", Rendiconti del Circolo Mathematico di Palermo 7:55–80.
- Harvey, F. Reese (1990). Spinors and Calibrations. San Diego: Academic Press. ISBN 0-12-329650-1.
- Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in Symmetries in Complex Analysis by Bruce Gilligan & Guy Roos, volume 468 of Contemporary Mathematics, American Mathematical Society.
- Schafer, Richard D. (1995). An introduction to non-associative algebras. Dover Publications. pp. 72–75. ISBN 0-486-68813-5.
- Springer, T. A.; F. D. Veldkamp (2000). Octonions, Jordan Algebras and Exceptional Groups. Springer-Verlag. ISBN 3-540-66337-1.
