1. Introduction
Let 
be an algebra over a field F not of characteristic two. The associator is a trilinear mapping
of 
into 
that measures the lack of associativity in
.
One scheme of classifying nonassociative algebras involves placing conditions on the associator of certain sets of elements. Some of the better known algebras are:
1) Alternative algebras. In this variety of algebras, all elements x and y satisfy
for all elements x and y. The octonion division ring is an alternative algebra. An interesting variation is psuedooctonion algebra (Okubo [5,9]).
2) Jordan algebras. These are commutative algebras in which all x and y satisfy
A Type D Jordan algebra is the Jordan algebra of the symmetric bilinear form q on a vector space
. Albert [3] has shown that any algebra of Type D has a basis 
with multiplication given by
The algebra will be semisimple if 
for all
.
3) Noncommutative Jordan algebras. A generalization of the alternative and Jordan algebras that requires all x and y satisfy a generalization of the commutative law
that is, the algebras are flexible, and
The book by Zhevlakov, Slin’ko, Shestakov and Shirshov [10] provides a detailed analysis of the alternative and Jordan rings.
The above algebras are all power associative since each element a generates an associative subalgebra; equivalently, 
for positive integers
. In any power associative algebra 
with unit element we can introduce the series
for 
ignoring the question of convergence.
An algebra 
over a field F is called quadratic if, for every x in 
where 
are in F and e is the identity of
. The quantities 
and 
are called the trace and norm of the element x, respectively. The trace is a linear functional on 
see Schafer [7]. The norm 
defines a symmetric bilinear form 
on 
via
Say 
is nondegenerate if 
is. Any quadratic algebra is power associative and any flexible, quadratic algebra is a noncommutative Jordan algebra.
A quadratic algebra 
is flexible if and only if the trace is associative; that is, 
for all 
in
. If 
is flexible then the mapping 
is an involution in 
(see Braun and Koecher [11], p. 216).
Lemma 1. The Hamiltonian division ring is a quadratic algebra.
Proof. Let 
be an element of the Hamitonian division ring. Direct computation shows that
.
Example 1. The octonion division ring is a quadratic algebras.
Example 2. Domokos and Kövesi-Domokos [12] propose a quadratic algebra, the “algebra of color” as a candidate for the algebra obeyed by a quantized field describing quarks and leptons (see also Wene [13,14], and Schafer [15]).
2. Construction of the Algebras
The elements of the algebra 
are the elements of the real vector space with basis
. The addition is the vector space addition and multiplication is defined by
, 
, 
is the identity and the distributive laws. We note that the algebra is commutative and has divisors of zero.
An immediate generalization of this algebra has a basis
, 
over the field 
of real numbers and multiplication defined by 
where 
is the identity. For want of a better name called these the Abbas algebras. As noted above, these algebras are Type D Jordan algebras. Note that the 
algebra is the construction for
; the results for the Abbas algebras apply to the
. Each Abbas algebra contains a copy of the complex numbers.
Lemma 2. The Abbas algebras are quadratic algebras.
Proof. Let H denote a Abbas algebra. Then if
, 
, Einstein summation convention where 
Then
Adding both sides gives
and we see that 
and
.
A commutative quadratic algebra will be a Jordan algebra.
Since the algebra is commutative the trace is associative; the norm is symmetric.
Lemma 3. The norm of a Abbas algebra is nondegenerate.
Proof. Let H denote a Abbas algebra. Then if
, 
is arbitrary and 
is fixed, then
3. Special 
Algebras
Hamieh and Abbas [1] pass to a representation of the point 
of the algebra 
in spherical coordinates, 
and 
. The subalgebras, called special 
algebras and denoted by 
are the subalgebras spanned by all elements in which the “azimutal phase angle 
is constant”. Each of these subalgebras is (isomorphic to) the complex numbers.
Lemma 4. The algebra 
is isomorphic to an algebra of two by two matrices
under the usual matrix operation of addition and multiplication.
Proof. The straight forward verification that the mapping 
is an isomorphism is left to the reader.
Lemma 5. Each of the algebras 
is isomorphic to the complex numbers.
Proof. We note that if 
then
if
. If 
or
, then 
and the subalgebra 
is (isomorphic to) the complex numbers. Otherwise, 
or 
for some
. Let
, then
The multiplication, using the basis 
will be given by
4. The Spinor Matrices
The classical reference on spinors and wave equations is the book by Corson [16].
The associator spinor matrices are
, 
, 
, 
where
. Denoting the 2 by 2 identity matrix by
, these matrices satisfy
The spinor matrices generate a 6-dimensional real algebra with elements
that contains the matrix representation of the 
algebra. Denote this algebra by
.
Lemma 6. The algebra 
is a quadratic algebra.
Proof. If 
is an element of
, then
Adding the left and right sides gives
Lemma 7. The algebra 
is flexible.
Proof. Because of the trilinearlty of the associator, we can write the elements x and y of the associator 
as 
and
. Then
Theorem 1. The algebra 
is a quadratic noncommutative Jordan algebra.
5. The Dirac Equation
We proceed as in Hamieh and Abbas [1]. The Dirac equation over the complex numbers is often written as
utilizing the Einstein summation convention for 
. A more general form is, setting
where
.
Upon substituting the matrices for 
and simplifying we get
In dimensions x and t, the solution is given by
p and 
are respect the momentum and energy. N is a normalization factor.
6. Conclusion
We have shown that the 
algebra belongs to a large class of Jordan algebras and have examined a few of the algebraic properties of these algebras and, like the Jordan algebra and the algebra of color, there is a very rich mathematical structure to further explore.
NOTES