We also say permits composition or that it obeys the composition law.
This definition is geometric in that quadratic forms give rise to geometric atributes for a vector space such as length, distance and orthogonality. Indeed, originally created over the real numbers such properties seem appropriate for an algebra; however, concepts of length and distance are less appropriate over arbitrary fields and encourage a second equivalent definition based solely on the algebraic aspect of such algebras.
The first definition makes the composition property part of the definition but obscures the alternative multiplication as well as the existence of an involutary anti-isomorphism for the algebra. The second definition makes both of these properties evident but obscures the composition property of the norm, and also hides the property that is a quadratic form. However both definitions have merit, the first captures the classical view of an algebra respecting a certain geometric condition while the second, introduced by Jacobson, promotes a purely algebraic treatment. In our examples and constructions to follows we attempt to exhibit both aspects by supplying the norm, the involution, and the product.
These strongly limit the structure of composition algebras and leads to the celebrated theorem of Hurwitz (see Theorem 4) which suitably classifies the composition algebras over . The work of many others including Albert, Dickson, Jacobson, and Kaplansky extended the essential conclussion of Hurwitz to all fields and the resulting generalization is still refered to as Hurwitz’s theorem.
There are other algebras with norms which permit composition in the sense that . For example, alternative algebras with involutions. However, the distinguishing property of composition algebras is that is a quadratic form. Classifications for such norms have been caried out by Schafer and McCrimmon.
Originally, composition algebras were created over the real numbers . Here the usual positive definite norm on the real vector space was used instead of the quadratic form (the square of the norm is the quadratic form).
The first non-trivial example is the set of complex numbers with where is assigned:
More interesting is the non-commutative algebra of Hamiltonians , created by Hamilton, where each has the form and
Because general fields do not sufficient squareroots, the use of norms in the classical Euclidean sense is replaced by the use of quadratic forms. Furthermore, the lack of ordering a field, such as a finite field, introduces the need to use non-degenerate rather than positive definite conditions. Under these generalizations composition algebras can be redefined form the classical context of composition algebras over to general composition algebras over arbitrary fields, as done by our original definitions above. In this context, there are three further composition algebras over .
Let with for all . Then is a composition algebra.
To check that has the compositional property let . Then
Note also that by defining then and . ∎
Let with for all . Then is a composition algebra.
Let and . Then . It is also evident that if then setting makes and also , where is the trace of . Hence
Therefore, . Since , it follows that is a symmetric bilinear form and so is quadratic form.
Finally, for composition note
Therefore is a composition algebra. ∎
This gives two new composition algebras over and indeed there is a third, constructed below as the algebra , which is 8-dimensional and non-associative but unlike the octonions, it has non-trivial zero-divisors.
A composition algebra is split if the quadratic form is isotropic.
The example of and just given are both examples of split composition algebras.
Immediately it follows that: for all ,
4 Constructing composition algebras
All of the following are composition algebras. [5, III.4]
, with trivial involution for all in .
For any , a quadratic extension of , that is
Here is a basis and has an involution defined by and .
For any , a quaternion algebra over defined as
Then forms a basis.11It is common to use for , but here is used exclusively for the underlying field. An involution is defined by , , and extended linearly.
For any , an octonion algebra over :
The set is a basis. An involution is defined by , , , and extended linearly.
Each of these algebras can be realized by the Cayley-Dickson method which takes an associative -algebra with involution and produces for each a new algebra on the vector space with product
Set the involution on to be .
The algebras are equipped with a trace , and norm . This norm serves as the quadratic map to establish these algebras a composition algebras. The images of the trace and norm lie in .
The new algebra is associative only if is commutative, otherwise it is alternative. This means that are the associative composition algebras.
5 Classification theorem
[2, Theorem 6.2.3] A composition algebra over a field with quadratic form is isomorphic to one of the following:
with trivial involution, so ,
Quadratic composition algebra: for ,
Quaternion algebra: for ,
Octonion algebra: for .
In particular, all composition algebras over , save perhaps those of type , are finite dimensional and of dimension , , or .
- 1 T.Y. Lam: Introduction to Quadratic Forms over Fields, AMS, Providence (2004).
- 2 N. Jacobson Structure theory of Jordan algebras, The University of Arkansas lecture notes in mathematics, vol. 5, Fayetteville, 1981.
- 3 K. McCrimmon: A Taste of Jordan Algebras, Springer, New York (2004).
- 4 J.H. Conway, D.A. Smith: On Quaternions and Octonions, Their Geometry, Arithmetic, and Symmetry, AK Peters, Natick, Mass (2003)
- 5 Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York, 1966.
|Date of creation||2013-03-22 15:11:33|
|Last modified on||2013-03-22 15:11:33|
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