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.Template:Rp If the ground field Template:Math is the field of real numbers and Template:Mvar is positive-definite, then Template:Mvar is called a Euclidean Hurwitz algebra.

Scalar product

If Template:Mvar has characteristic not equal to 2, then a bilinear form Template:Math is associated with the quadratic form Template:Mvar.

Involution in Hurwitz algebras

Assuming Template:Mvar has a multiplicative unity, define involution and right and left multiplication operators by

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

Evidently Template: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. Template:Math
• Template:Math
• Template:Math, Template:Math, where Template:Math denotes the adjoint operator with respect to the form Template:Math
• Template:Math where Template:Math
• Template:Math
• Template:Math, Template:Math, so that Template:Mvar is an alternative algebra

These properties are proved starting from polarized version of the identity Template:Math:

$2(a,b)(c,d)=(ac,bd)+(ad,bc).$

Setting Template:Math or Template:Math yields Template:Math and Template:Math. Hence Template:Math. Similarly Template:Math. Hence Template:Math. By the polarized identity Template:Math so Template:Math. Applied to 1 this gives Template:Math. Replacing Template:Mvar by Template:Overline gives the other identity. Substituting the formula for Template:Math in Template:Math gives Template:Math.

Para-Hurwitz algebra

Another operation Template:Math may be defined in a Hurwitz algebra as

Template:Math

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

A para-Hurwitz algebra satisfiesTemplate:Rp

$(x*y)*x=x*(y*x)=⟨x|x⟩y.$

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.Template:Rp Similarly, a flexible algebra satisfying

$⟨xy|xy⟩=⟨x|x⟩⟨y|y⟩$

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

References

• Template:Eom
• Template:Citation
• Susumu Okubo & J. Marshall Osborn (1981) "Algebras with nondegenerate associative symmetric bilinear forms permitting composition", Communications in Algebra 9(12): 1233–61, Template:Mr and 9(20): 2015–73 Template:Mr.