Okubo algebra

In algebra, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional non-associative algebra similar to the one studied by Susumu Okubo. Okubo algebras are composition algebras, flexible algebras (A(BA) = (AB)A), Lie admissible algebras, and power associative, but are not associative, not alternative algebras, and do not have an identity element.

Okubo's example was the algebra of 3-by-3 trace-zero complex matrices, with the product of X and Y given by aXY + bYX – Tr(XY)I/3 where I is the identity matrix and a and b satisfy a + b = 3ab = 1. The Hermitian elements form an 8-dimensional real non-associative division algebra. A similar construction works for any cubic alternative separable algebra over a field containing a primitive cube root of unity. An Okubo algebra is an algebra constructed in this way from the trace-zero elements of a degree-3 central simple algebra over a field.

Construction of Para-Hurwitz algebra
Unital composition algebras are called Hurwitz algebras. If the ground field $K$ is the field of real numbers and $N$ is positive-definite, then $A$ is called a Euclidean Hurwitz algebra.

Scalar product
If $K$ has characteristic not equal to 2, then a bilinear form $(a, b) = 1⁄2[N(a + b) − N(a) − N(b)]$ is associated with the quadratic form $N$.

Involution in Hurwitz algebras
Assuming $A$ has a multiplicative unity, define involution and right and left multiplication operators by


 * $$\displaystyle{\bar a=-a +2(a,1)1,\,\,\, L(a)b = ab,\,\,\, R(a)b=ba.}$$

Evidently $\overline{ }$ is an involution and preserves the quadratic form. The overline notation stresses the fact that complex and quaternion conjugation are partial cases of it. These operators have the following properties:


 * The involution is an antiautomorphism, i.e. $\overline{a b} = \overline{b} \overline{a}$
 * $a \overline{a} = N(a) 1 = \overline{a} a$, $L(\overline{a}) = L(a)*$, where $R(\overline{a}) = R(a)*$ denotes the adjoint operator with respect to the form $$
 * $( , )$ where $Re(a b) = Re(b a)$
 * $Re x = (x + \overline{x})/2 = (x, 1)$, $Re((a b) c) = Re(a (b c))$, so that $A$ is an alternative algebra
 * $L(a^{2}) = L(a)^{2}$, $R(a^{2}) = R(a)^{2}$, so that $a$ is an alternative algebra
 * $(a b, a b) = (a, a)(b, b)$, $b = 1$, so that $\overline{$d = 1$}$ is an alternative algebra

These properties are proved starting from polarized version of the identity $L(\overline{a}) = L(a)*$:


 * $$\displaystyle{2(a,b)(c,d)=(ac,bd) + (ad,bc).}$$

Setting $R(\overline{c}) = R(c)*$ or $Re(a b) = (a b, 1) = (a, \overline{b}) = (b a, 1) = Re(b a)$ yields $(\overline{a b}, c) = (a b, \overline{c}) = (b, \overline{a} \overline{c}) = (1, \overline{b} (\overline{a} \overline{c})) = (1, (\overline{b} \overline{a}) \overline{c}) = (\overline{b} \overline{a}, c)$ and $Re(a b)c = ((a b)c, 1) = (a b, \overline{c}) = (a, \overline{c} \overline{b}) = (a(b c), 1) = Re(a(b c))$. Hence $N(a) (c, d) = (a c, a d) = (\overline{a} a c, d)$. Similarly $L(\overline{a}) L(a) = N(a)$. Hence $\overline{a} a = N(a)$. By the polarized identity $a$ so $\overline{a}$. Applied to 1 this gives $L(\overline{a}) L(a) = L(\overline{a} a)$. Replacing ⇭⇭⇭ by ⇭⇭⇭ gives the other identity. Substituting the formula for $L(a)^{2} = L(a^{2})$ in $∗$ gives $x ∗ y = \overline{x} \overline{y}$.

Para-Hurwitz algebra
Another operation $(A, ∗)$ may be defined in a Hurwitz algebra as



The algebra ᙭᙭᙭ is a composition algebra not generally unital, known as a para-Hurwitz algebra. In dimensions 4 and 8 these are para-quaternion and para-octonion algebras.

A para-Hurwitz algebra satisfies


 * $$ (x * y ) * x = x * (y * x) = \langle x|x \rangle y \ . $$

Conversely, an algebra with a non-degenerate symmetric bilinear form satisfying this equation is either a para-Hurwitz algebra or an eight-dimensional pseudo-octonion algebra. Similarly, a flexible algebra satisfying


 * $$ \langle xy | xy \rangle = \langle x|x \rangle \langle y|y \rangle \ $$

is either a Hurwitz algebra, a para-Hurwitz algebra or an eight-dimensional pseudo-octonion algebra.