This is the definition of alternative algebra; according to Hurwitz theorem , only four exist which are R real numbers, C complex numbers ~ U(1), H quaternions ~ SU(2), and O octonions
This second edition contains additional exercises, plus new student projects on skewfields, quaternions, and octonions
Many new exercises have been added, while new study projects examine skewfields, quaternions, and octonions
Aside from their obvious connection to the theories of Brauer groups and module categories over monoidal categories , it seems also reasonable to expect that such division algebras would have similar applications in quantum (specifically, nonassociative) geometry and physics as their classical analogue quaternions, octonions
and generalizations do, see e.
The resulting algebra is again a division algebra, the octonions
O, constructed independently by Graves in 1843 and Cayley in 1845.
8] of 3 x 3 Hermitian matrices over the Octonions
is simple with Pierce constant d = 8 but does not admit a representation.
The importance of these identities is that they are intimately connected to certain division algebras, namely the ones for the reals, complex numbers, quaternions, and octonions
, corresponding to dimension n = 1, 2, 4, 8, .
The algebra of octonions
[1-3] is interesting mathematical structure for physical applications (see reviews [4-7]).
Their topics include affine and conformal transformations of rational Bezier curves, the G2-congruence classes of curves in the purely imaginary octonions
, surfaces in the four-dimensional Euclidean and Minkowski space, radial transversal light-like hypersurfaces of almost complex manifolds with Norden metric, and the sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold.
are a bigger and better version of the complex numbers, but with some subtle properties, say Dray and Manogue: bigger because there are more square roots of -1, better because octonionic formalism provides natural explanations for several intriguing results in both mathematics and physics, and subtle because the rules are more complicated--order matters.
The first set of 120 quaternions can be expressed as 120 special unit quaternions known as icosians which telescope 4-D discrete-space quaternions up to being 8-D discrete-space octonions
to locate points that form a special lattice in [R.
These special 120 icosians are to be considered as special octonions
, 8-tuples of rational numbers which, with respect to a particular norm, form part of a special lattice in [R.