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'' = 3''ab'' = 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.Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol (1998) "Composition and Triality", chapter 8 in ''The Book of Involutions'', pp 451–511, Colloquium Publications v 44, American Mathematical Society

Construction of Para-Hurwitz algebra

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

Scalar product

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

Involution in Hurwitz algebras

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

:$\displaystyle$

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:

* The involution is an antiautomorphism, i.e.
*
* , , where denotes the adjoint operator with respect to the form
* where
*
* , , so that is an alternative algebra

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

:$\displaystyle$

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

Para-Hurwitz algebra

Another operation 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-quaternionThe term "para-quaternions" is sometimes applied to unrelated algebras. 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.

References

*
*
* Susumu Okubo & J. Marshall Osborn (1981) "Algebras with nondegenerate associative symmetric bilinear forms permitting composition", Communications in Algebra 9(12): 1233–61, and 9(20): 2015–73 {{mr, id=0640611.

Composition algebras
Non-associative algebras