Associative Composition Algebra/Bibinarions

A construction of a doubled algebra was initiated by L. E. Dickson and recounted by A. A. Albert. The method presumes an algebra with conjugate * and produces one of double the dimension and a new conjugation: (u, v)* = (u*, −v) where u* denotes the original conjugation. The new algebra has products given by

$(u,v)\times (w,x)=(uw-vx^{*},ux+vw^{*}).$

Starting with a field and its conjugation, a sequence of algebras can be so constructed. The division binarions arises when the starting field is the real numbers $\mathbb {R} .$ Since the reals have no conjugation, the identity is substituted, and the conjugation arises as above. Continuing with the construction, using the binarion conjugate, quaternions are obtained as will be seen. However, the binarion conjugation may be forgotten (identity substituted), and bibinarions produced according to the Dickson/Albert method given above.

A bibinarion is then a pair of division binarions (u, v) with conjugation (u, v)* = (u, −v). The norm of a bibinarion is then

$(u,v)(u,v)^{*}=(u,v)(u,-v)=(u^{2}-v(-v),\ u(-v)+vu)=(u^{2}+v^{2},\ 0).$

Notice that this norm is a division binarion, and is not the kind of norm that produces a metric.

Furthermore, with i² = −1 in C, a bibinarion (u, i u) has zero norm. Such an element is called a null vector. The bibinarions form a split algebra since some elements are null vectors.

The product of two bibinarions is commutative since the generating conjugation is the identity. Most remarkably, there is bibinarion j = (0, i) with j² = (0, i)² = (−i², 0) = +1. The two-dimensional subalgebra of bibinarions on basis {1, j } is called split binarions.

Let u = i sin θ and v = i cos θ. Then for any θ the norm of (u,v) is

$-\sin ^{2}\theta -\cos ^{2}\theta =-1.$ The "anti-circle" can be formed by {(u,v) : 0 ≤ θ < 2 π }, a circle with imaginary radius in the space of bibinarions.

History

The idea of an algebra with two imaginary units that commute was considered in mid-19th century Britain. Hamilton used a commuting h with his biquaternions. James Cockle saw that the square of the product hi of imaginary units was plus one, thus creating "a new imaginary in algebra" as he wrote in Philosophical Magazine in 1848. His use of the letter j,   j² = +1, has been widely adopted. Although Hamilton provided a vocabulary of vector operations (including the del operator), these explorations preceded set theory, group theory, and the unfolding of mathematical notation. With 1 on the real axis, the two imaginary units h and i, and their product hi, Cockle's commutative algebra T (tessarines) has a real basis of four elements. By the end of the 19th century tessarines and quaternions were referred to as hypercomplex numbers.

In 1892 Corrado Segre introduced bicomplex numbers in Mathematische Annalen (v 40: 455 to 67).

The division binarion basis of this algebra is used in the Dickson-style construction of biquaternions.