octonion造句

• The algebra of bioctonions is an example of an octonion algebra.
• But like the octonion product it is not uniquely defined.
• In particular, this gives the classification of octonion algebras.
• Moreover, the Moufang identities hold in any octonion algebra.
• The Cayley plane uses octonion coordinates which do not satisfy the associative law.
• There are corresponding split octonion algebras over any field " F ".
• Instead there are many different cross products, each one dependent on the choice of octonion product.
• This property is shared by most binary operations, but not subtraction or division or octonion multiplication.
• This one is derived from its parent octonion ( one of 480 possible ), which is defined by:
• The algebra of bioctonions is the octonion algebra over the complex numbers "'C " '.
• The above definition though is not unique, but is only one of 480 possible definitions for octonion multiplication with.
• A convenient mnemonic is given by the diagram at the right which represents the multiplication table for the split octonion.
• Up to isomorphism, the octonions and the split-octonions are the only two octonion algebras over the real numbers.
• Up to "'R "'- algebra isomorphism, these are the only octonion algebras over the reals.
• It follows that the invertible elements in any octonion algebra form a Moufang loop, as do the elements of unit norm.
• The irreducible integral octonions are exactly those of prime norm, and every integral octonion can be written as a product of irreducible octonions.
• This group is related to the octonions by considering the 16 component spinors as two component octonion spinors and the gamma matrices acting on the upper indices as unit octonions.
• 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.
• An algebraic symmetry is lost with each increase in dimensionality : quaternion multiplication is not commutative, octonion multiplication is non-associative, and the norm of sedenions is not multiplicative.
• Note that in keeping associative and thus not reducing the 4-dimensional geometric algebra to an octonion one, the whole multiplication table can be derived from the equation for  ".