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 have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension.
They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative.
They are also power associative.
Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used.
Octonions are related to exceptional structures in mathematics, among them the exceptional Lie groups.
Octonions have applications in fields such as string theory, special relativity and quantum logic.
Applying the Cayley–Dickson construction to the octonions produces the sedenions.
History
The octonions were discovered in 1843 by John T. Graves, inspired by his friend William Rowan Hamilton's discovery of quaternions.
Graves called his discovery "octaves", and mentioned them in a letter to Hamilton dated 26 December 1843.Sabadini, Irene, et al. “p 168.”
Hypercomplex Analysis, Birkhäuser, Basel, 2009, https://books.google.com/books?id=H-5v6pPpyb4C&lpg=PA168&ots=GBaIDVCCtS&dq=december%2026%2C%201843%20octonion&pg=PA168#v=onepage&q=december%2026,%201843%20octonion&f=false.
He first published his result  slightly later than Arthur Cayley's article.
The octonions were discovered independently by Cayley.
Appendix reprinted in The Collected Mathematical Papers, Johnson Reprint Co., New York, 1963, p. 127 and are sometimes referred to as "Cayley numbers" or the "Cayley algebra".
Hamilton described the early history of Graves' discovery.
Definition
The octonions can be thought of as octets (or 8-tuples) of real numbers.
Every octonion is a real linear combination of the unit octonions:
\{e_0, e_1, e_2, e_3, e_4, e_5, e_6, e_7\},
where  is the scalar or real element; it may be identified with the real number 1.
That is, every octonion  can be written in the form
x = x_0e_0 + x_1e_1 + x_2e_2 + x_3e_3 + x_4e_4 + x_5e_5 + x_6e_6 + x_7e_7,\,
with real coefficients .
Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions.
Multiplication is more complex.
Multiplication is distributive over addition, so the product of two octonions can be calculated by summing the products of all the terms, again like quaternions.
The product of each pair of terms can be given by multiplication of the coefficients and a multiplication table of the unit octonions, like this one (due to Cayley, 1845, and Graves, 1843):
Most off-diagonal elements of the table are antisymmetric, making it almost a skew-symmetric matrix except for the elements on the main diagonal, as well as the row and column for which  is an operand.
The table can be summarized as follows:
e_i e_j  = \begin{cases} e_j , & \text{if }i = 0 \\ e_i , & \text{if }j = 0 \\ - \delta_{ij}e_0 + \varepsilon _{ijk} e_k, & \text{otherwise} \end{cases}
where  is the Kronecker delta (equal to 1 if and only if ), and  is a completely antisymmetric tensor with value 1 when .
The above definition is not unique however; it is only one of 480 possible definitions for octonion multiplication with .
The others can be obtained by permuting and changing the signs of the non-scalar basis elements .
The 480 different algebras are isomorphic, and there is rarely a need to consider which particular multiplication rule is used.
Each of these 480 definitions is invariant up to signs under some 7-cycle of the points (1234567), and for each 7-cycle there are four definitions, differing by signs and reversal of order.
A common choice is to use the definition invariant under the 7-cycle (1234567) with  — by using the triangular multiplication diagram, or Fano plane below that also shows the sorted list of 124 based 7-cycle triads and its associated multiplication matrices in both  and IJKL format.
1000px|Octonion triads, Fano plane, and multiplication matrices
A variation of this sometimes used is to label the elements of the basis by the elements , 0, 1, 2, ..., 6, of the projective line over the finite field of order 7.
The multiplication is then given by  and , and all expressions obtained from this by adding a constant (modulo 7) to all subscripts: in other words using the seven triples (124) (235) (346) (450) (561) (602) (013).
These are the nonzero codewords of the quadratic residue code of length 7 over the Galois field of two elements, .
There is a symmetry of order 7 given by adding a constant mod 7 to all subscripts, and also a symmetry of order 3 given by multiplying all subscripts by one of the quadratic residues 1, 2, 4 mod 7..
Available as ArXive preprint Figure 1 is located here.
The multiplication table for a geometric algebra of signature  can be given in terms of the following 7 quaternionic triples (omitting the identity element):
in which the lowercase items are vectors and the uppercase ones are bivectors and  (which is the Hodge star operator).
If the  is forced to be equal to the identity then the multiplication ceases to be associative, but the  may be removed from the multiplication table resulting in an octonion multiplication table.
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 .
Consider the gamma matrices.
The formula defining the fifth gamma matrix shows that it is the  of a four-dimensional geometric algebra of the gamma matrices.
Cayley–Dickson construction
A more systematic way of defining the octonions is via the Cayley–Dickson construction.
Just as quaternions can be defined as pairs of complex numbers, the octonions can be defined as pairs of quaternions.
Addition is defined pairwise.
The product of two pairs of quaternions  and  is defined by
(a,b)(c,d)=(ac-d^{*}b,da+bc^{*}),
where  denotes the conjugate of the quaternion .
This definition is equivalent to the one given above when the eight unit octonions are identified with the pairs
Fano plane mnemonic
thumb|A mnemonic for the products of the unit octonions.
A convenient mnemonic for remembering the products of unit octonions is given by the diagram, which represents the multiplication table of Cayley and Graves.
Figure 29.1: Representation of multiplication table on projective plane.
This diagram with seven points and seven lines (the circle through 1, 2, and 3 is considered a line) is called the Fano plane.
The lines are directional.
The seven points correspond to the seven standard basis elements of  (see definition below).
Each pair of distinct points lies on a unique line and each line runs through exactly three points.
Let  be an ordered triple of points lying on a given line with the order specified by the direction of the arrow.
Then multiplication is given by
and
together with cyclic permutations.
These rules together with
1 is the multiplicative identity,
for each point in the diagram
completely defines the multiplicative structure of the octonions.
Each of the seven lines generates a subalgebra of  isomorphic to the quaternions .
Conjugate, norm, and inverse
The conjugate of an octonion
x = x_0\,e_0 + x_1\,e_1 + x_2\,e_2 + x_3\,e_3 + x_4\,e_4 + x_5\,e_5 + x_6\,e_6 + x_7\,e_7
is given by
x^* = x_0\,e_0 - x_1\,e_1 - x_2\,e_2 - x_3\,e_3 - x_4\,e_4 - x_5\,e_5 - x_6\,e_6 - x_7\,e_7.
Conjugation is an involution of  and satisfies  (note the change in order).
The real part of  is given by
\frac{x + x^*}{2} = x_0\,e_0
and the imaginary part by
\frac{x - x^*}{2} = x_1\,e_1 + x_2\,e_2 + x_3\,e_3 + x_4\,e_4 + x_5\,e_5 + x_6\,e_6 + x_7\,e_7.
The set of all purely imaginary octonions span a 7-dimensional subspace of , denoted .
Conjugation of octonions satisfies the equation
x^* =-\tfrac{1}{6} \bigl(x+(e_1x)e_1+(e_2x)e_2+(e_3x)e_3+(e_4x)e_4+(e_5x)e_5+(e_6x)e_6+(e_7x)e_7\bigr).
The product of an octonion with its conjugate, , is always a nonnegative real number:
x^*x = x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2.
Using this the norm of an octonion can be defined, as
\|x\| = \sqrt{x^*x}.
This norm agrees with the standard 8-dimensional Euclidean norm on .
The existence of a norm on  implies the existence of inverses for every nonzero element of .
The inverse of , which is the unique octonion  satisfying , is given by
x^{-1} = \frac {x^*}{\|x\|^2}.
Properties
Octonionic multiplication is neither commutative:
if ,  are distinct and non-zero,
nor associative:
if , ,  are distinct, non-zero and .
The octonions do satisfy a weaker form of associativity: they are alternative.
This means that the subalgebra generated by any two elements is associative.
Actually, one can show that the subalgebra generated by any two elements of  is isomorphic to , , or , all of which are associative.
Because of their non-associativity, octonions cannot be represented by a subalgebra of a matrix ring over \mathbb R, unlike the real numbers, complex numbers and quaternions.
The octonions do retain one important property shared by , , and : the norm on  satisfies
\|xy\| = \|x\|\|y\|.
This equation means that the octonions form a composition algebra.
The higher-dimensional algebras defined by the Cayley–Dickson construction (starting with the sedenions) all fail to satisfy this property.
They all have zero divisors.
Wider number systems exist which have a multiplicative modulus (for example, 16-dimensional conic sedenions).
Their modulus is defined differently from their norm, and they also contain zero divisors.
As shown by Hurwitz, , , , and  are the only normed division algebras over the real numbers.
These four algebras also form the only alternative, finite-dimensional division algebras over the real numbers (up to isomorphism).
Not being associative, the nonzero elements of  do not form a group.
They do, however, form a loop, specifically a Moufang loop.
Commutator and cross product
The commutator of two octonions  and  is given by
[x, y] = xy - yx.
This is antisymmetric and imaginary.
If it is considered only as a product on the imaginary subspace  it defines a product on that space, the seven-dimensional cross product, given by
x \times y = \tfrac{1}{2}(xy - yx).
Like the cross product in three dimensions this is a vector orthogonal to  and  with magnitude
\|x \times y\| = \|x\| \|y\| \sin \theta.
But like the octonion product it is not uniquely defined.
Instead there are many different cross products, each one dependent on the choice of octonion product.
Automorphisms
An automorphism, , of the octonions is an invertible linear transformation of  which satisfies
A(xy) = A(x)A(y).
The set of all automorphisms of  forms a group called .
The group  is a simply connected, compact, real Lie group of dimension 14.
This group is the smallest of the exceptional Lie groups and is isomorphic to the subgroup of  that preserves any chosen particular vector in its 8-dimensional real spinor representation.
The group  is in turn a subgroup of the group of isotopies described below.
See also:  – the automorphism group of the Fano plane.
Isotopies
An isotopy of an algebra is a triple of bijective linear maps , ,  such that if  then .
For  this is the same as an automorphism.
The isotopy group of an algebra is the group of all isotopies, which contains the group of automorphisms as a subgroup.
The isotopy group of the octonions is the group , with , ,  acting as the three 8-dimensional representations.
The subgroup of elements where  fixes the identity is the subgroup , and the subgroup where , ,  all fix the identity is the automorphism group .
Applications
The octonions play a significant role in the classification and construction of other mathematical entities.
For example, the exceptional Lie group  is the automorphism group of the octonions, and the other exceptional Lie groups , ,  and  can be understood as the isometries of certain projective planes defined using the octonions.Baez (2002), section 4.
The set of self-adjoint 3 × 3 octonionic matrices, equipped with a symmetrized matrix product, defines the Albert algebra.
In discrete mathematics, the octonions provide an elementary derivation of the Leech lattice, and thus they are closely related to the sporadic simple groups.
Applications of the octonions to physics have largely been conjectural.
For example, in the 1970s, attempts were made to understand quarks by way of an octonionic Hilbert space.
<br/> It is known that the octonions, and the fact that only four normed division algebras can exist, relates to the spacetime dimensions in which supersymmetric quantum field theories can be constructed.
Also, attempts have been made to obtain the Standard Model of elementary particle physics from octonionic constructions, for example using the "Dixon algebra" .
<br /><br /><br /><br /><br />
Octonions have also arisen in the study of black hole entropy and quantum information science.
Octonions have been used in solutions to the hand eye calibration problem in robotics.Jin Wu, et al.
"Hand-eye Calibration: 4D Procrustes Analysis Approach." 6 August 2019.
Deep octonion networks provide a means of efficient and compact expression in machine learning applications.Wu, Jiasong, Ling Xu, Fuzhi Wu, Youyong Kong, Lotfi Senhadji, and Huazhong Shu.
"Deep octonion networks."
Neurocomputing (2020).
Integral octonions
There are several natural ways to choose an integral form of the octonions.
The simplest is just to take the octonions whose coordinates are integers.
This gives a nonassociative algebra over the integers called the Gravesian octonions.
However it is not a maximal order (in the sense of ring theory); there are exactly seven maximal orders containing it.
These seven maximal orders are all equivalent under automorphisms.
The phrase "integral octonions" usually refers to a fixed choice of one of these seven orders.
These maximal orders were constructed by , Dickson and Bruck as follows.
Label the eight basis vectors by the points of the projective line over the field with seven elements.
First form the "Kirmse integers" : these consist of octonions whose coordinates are integers or half integers, and that are half integers (that is, halves of odd integers) on one of the 16 sets
of the extended quadratic residue code of length 8 over the field of two elements, given by ,  and its images under adding a constant modulo 7, and the complements of these eight sets.
Then switch infinity and any one other coordinate; this operation creates a bijection of the Kirmse integers onto a different set, which is a maximal order.
There are seven ways to do this, giving seven maximal orders, which are all equivalent under cyclic permutations of the seven coordinates 0123456.
(Kirmse incorrectly claimed that the Kirmse integers also form a maximal order, so he thought there were eight maximal orders rather than seven, but as  pointed out they are not closed under multiplication; this mistake occurs in several published papers.)
The Kirmse integers and the seven maximal orders are all isometric to the  lattice rescaled by a factor of .
In particular there are 240 elements of minimum nonzero norm 1 in each of these orders, forming a Moufang loop of order 240.
The integral octonions have a "division with remainder" property: given integral octonions  and , we can find  and  with , where the remainder  has norm less than that of .
In the integral octonions, all left ideals and right ideals are 2-sided ideals, and the only 2-sided ideals are the principal ideals  where  is a non-negative integer.
The integral octonions have a version of factorization into primes, though it is not straightforward to state because the octonions are not associative so the product of octonions depends on the order in which one does the products.
The irreducible integral octonions are exactly those of prime norm, and every integral octonion can be written as a product of irreducible octonions.
More precisely an integral octonion of norm  can be written as a product of integral octonions of norms  and .
The automorphism group of the integral octonions is the group  of order 12,096, which has a simple subgroup of index 2 isomorphic to the unitary group .
The isotopy group of the integral octonions is the perfect double cover of the group of rotations of the  lattice.
See also
Octonion algebra
Okubo algebra
Split-octonions
Triality
Notes
References
.
(Review).
External links
Koutsoukou-Argyraki, Angeliki.
Octonions (Formal proof development in Isabelle/HOL, Archive of Formal Proofs)
