Grigoryan and Agaian provide the mathematics of quaternions and octonions
and show how to use them in emerging areas of color image processing.
Zeren, "Multifluid plasma equations in terms of hyperbolic octonions
," International Journal of Geometric Methods in Modern Physics, vol.
Our investigation arose in fact out of the question whether the nonassociative octonions
(more precisely, the 7-dimensional commutator algebra of imaginary octonions
) can be viewed as part of an [L.sub.[infinity]] algebra.
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
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.g.
The resulting algebra is again a division algebra, the octonions
O, constructed independently by Graves in 1843 and Cayley in 1845.
The Euclidean Jordan algebra [H.sub.3.sup.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, ...
Generalizing from rotation around one axis to rotations around three orthogonal axes (as in 3-dimensional space) leads Penrose from complex numbers to hyper-complex numbers (such as quaternions and octonions
), and to the hypercomplex algebras discovered by Clifford, Cayley, and Grassman when spaces of more dimensions (and arbitrary signatures) are considered.
And it inspired the subsequent scholars to apply the quaternions, octonions
, and sedenions to investigate the gravitational field, electromagnetic field, nuclear field, Dirac wave equation, Yang-Mills equation, electroweak field, color confinement, dark matter field, and so forth.
This second edition contains additional exercises, plus new student projects on skewfields, quaternions, and octonions
. ([umlaut] Ringgold, Inc., Portland, OR)