In
mathematics, a bioctonion, or complex octonion, is a pair (''p,q'') where ''p'' and ''q'' are
biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...
s.
The product of two bioctonions is defined using biquaternion multiplication and the biconjugate p → p*:
:
The bioctonion ''z'' = (''p,q'') has conjugate ''z''* = (''p''*, – ''q'').
Then norm ''N''(''z'') of bioctonion ''z'' is ''z z''* = ''p p''* + ''q q''*, which is a complex
quadratic form with eight terms.
The bioctonion algebra is sometimes introduced as simply the
complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...
of real
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s, but in
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
it is the result of the
Cayley–Dickson construction
In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced b ...
that begins with the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, the trivial involution, and quadratic form z
2. The algebra of bioctonions is an example of an
octonion algebra In mathematics, an octonion algebra or Cayley algebra over a field ''F'' is a composition algebra over ''F'' that has dimension 8 over ''F''. In other words, it is a unital non-associative algebra ''A'' over ''F'' with a non-degenerate quadratic ...
.
For any pair of bioctonions ''y'' and ''z'',
:
showing that ''N'' is a quadratic form admitting composition, and hence the bioctonions form a
composition algebra
In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies
:N(xy) = N(x)N(y)
for all and in .
A composition algebra includes an involution ...
.
Complex octonions have been used to describe the generations of
quarks and
leptons.
[C. Furey (2016]
Standard Model Physics from an Algebra ?
/ref>
References
* J. D. Edmonds (1978
Nine-vectors, complex octonion/quaternion hypercomplex numbers, Lie groups and the ‘real’ world
Foundations of Physics
''Foundations of Physics'' is a monthly journal "devoted to the conceptual bases and fundamental theories of modern physics and cosmology, emphasizing the logical, methodological, and philosophical premises of modern physical theories and procedur ...
8(3-4): 303–11, link from PhilPapers
PhilPapers is an interactive academic database of Academic journal, journal articles in philosophy. It is maintained by the Centre for Digital Philosophy at the University of Western Ontario, and as of 2022, it has "394,867 registered users, incl ...
.
* J. Koeplinger & V. Dzhunushaliev (2008
"Nonassociative decomposition of angular momentum operator using complex octonions"
presentation at a meeting of the American Physical Society
* D.G. Kabe (1984) "Hypercomplex Multivariate Normal Distribution", Metrika 31(2):63−76
* A.A. Eliovich & V.I. Sanyuk (2010) "Polynorms", ''Theoretical and Mathematics Physics'' 162(2) 135−48
{{Number systems
Composition algebras
Octonions