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.
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.
Assuming A has a multiplicative unity, define involution and right and left multiplication operators by
Evidently 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:
These properties are proved starting from polarized version of the identity (a b, a b) = (a, a)(b, b):
Setting b = 1 or d = 1 yields L(a) = L(a)* and R(c) = R(c)*. Hence Re(a b) = (a b, 1) = (a, b) = (b a, 1) = Re(b a). Similarly (a b, c) = (a b, c) = (b, a c) = (1, b (a c)) = (1, (b a) c) = (b a, c). Hence Re(a b)c = ((a b)c, 1) = (a b, c) = (a, c b) = (a(b c), 1) = Re(a(b c)). By the polarized identity N(a) (c, d) = (a c, a d) = (a a c, d) so L(a) L(a) = N(a). Applied to 1 this gives a a = N(a). Replacing a by a gives the other identity. Substituting the formula for a in L(a) L(a) = L(a a) gives L(a)2 = L(a2).
Another operation ∗ may be defined in a Hurwitz algebra as
A para-Hurwitz algebra satisfies: 48
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.: 49 Similarly, a flexible algebra satisfying
is either a Hurwitz algebra, a para-Hurwitz algebra or an eight-dimensional pseudo-octonion algebra.