| Octonions | |
|---|---|
| Symbol | |
| Type | Hypercomplexalgebra |
| Units | e0, ..., e7 |
| Multiplicative identity | e0 |
| Main properties | |
| Common systems | |
Less common systems | |
Inmathematics, theoctonions are anormed division algebra over thereal numbers, a kind ofhypercomplexnumber system. The octonions are usually represented by the capital letter O, using boldfaceO orblackboard bold. Octonions have eightdimensions; twice the number of dimensions of thequaternions, of which they are an extension. They arenoncommutative andnonassociative, but satisfy a weaker form of associativity; namely, they arealternative. They are alsopower associative.
Octonions are not as well known as the quaternions andcomplex numbers, which are much more widely studied and used. Octonions are related to exceptional structures[clarification needed] in mathematics, among them theexceptional Lie groups. Octonions have applications in fields such asstring theory,special relativity andquantum logic. Applying theCayley–Dickson construction to the octonions produces thesedenions.
The octonions were discovered in December 1843 byJohn T. Graves, inspired by his friendWilliam Rowan Hamilton's discovery of quaternions. Shortly before Graves' discovery of octonions, Graves wrote in a letter addressed to Hamilton on October 26, 1843, "If with your alchemy you can make three pounds of gold, why should you stop there?"[1]
Graves called his discovery "octaves", and mentioned them in a letter to Hamilton dated 26 December 1843.[2] He first published his result slightly later thanArthur Cayley's article.[3] The octonions were discovered independently by Cayley[4] and are sometimes referred to asCayley numbers or theCayley algebra. Hamilton described the early history of Graves's discovery.[5]
The octonions can be thought of as octets (or 8-tuples) of real numbers. Every octonion is a reallinear combination of theunit octonions:
wheree0 is the scalar or real element; it may be identified with the real number1 . That is, every octonionx can be written in the form
with real coefficientsxi.
A more systematic way of defining the octonions is via the Cayley–Dickson construction. Applying the Cayley–Dickson construction to the quaternions produces the octonions, which can be expressed as.[6]
Much 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(a,b) and(c,d) is defined by
wherez* denotes theconjugate of the quaternionz. This definition is equivalent to the one given above when the eight unit octonions are identified with the pairs
Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions.
Multiplication of octonions is more complex. Multiplication isdistributive 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 amultiplication table of the unit octonions, like this one (given both byArthur Cayley in 1845 andJohn T. Graves in 1843):[7]
Most off-diagonal elements of the table are antisymmetric, making it almost askew-symmetric matrix except for the elements on the main diagonal, as well as the row and column for whiche0 is an operand.
The table can be summarized as follows:[8]
whereδℓm is theKronecker delta (equal to1 ifℓ =m, and0 forℓ ≠m), andεℓmn is acompletely antisymmetric tensor with value+1 whenℓ m n = 1 2 3, 1 4 5, 1 7 6, 2 4 6, 2 5 7, 3 4 7, 3 6 5 , and any even number ofpermutations of the indices, but−1 for any oddpermutations of the listed triples (e.g. but however, again). Whenever any two of the three indices are the same,εℓmn= 0 .
The above definition is not unique, however; it is only one of 480 possible definitions for octonion multiplication withe0 = 1. The others can be obtained by permuting and changing the signs of the non-scalar basis elements{e1,e2,e3,e4,e5,e6,e7} . The 480 different algebras areisomorphic, 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 (1 2 3 4 5 6 7) , 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) withe1e2 =e4 by using the triangular multiplication diagram, or Fano plane below that also shows the sorted list of1 2 4 based 7-cycle triads and its associated multiplication matrices in bothen and format.
A variant of this sometimes used is to label the elements of the basis by the elements∞, 0, 1, 2, ..., 6, of theprojective line over thefinite field of order 7. The multiplication is then given bye∞ = 1 ande0e1 =e3, and all equations obtained from this one by adding a constant (modulo 7) to all subscripts: In other words using the seven triples(0 1 3),(1 2 4),(2 3 5),(3 4 6),(4 5 0),( 5 6 1),(6 0 2) . These are the nonzero codewords of thequadratic residue code of length 7 over theGalois field of two elements,GF(2). There is a symmetry of order 7 given by adding a constantmod 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 .[9][10]These seven triples can also be considered as the seven translates of the set {1,2,4} of non-zero squares forming a cyclic (7,3,1)-difference set in the finite fieldGF(7) of seven elements.
The Fano plane shown above with and IJKL multiplication matrices also includes thegeometric algebra basis with signature(− − − −) and is given in terms of the following 7 quaternionic triples (omitting the scalar identity element):
or alternatively:
in which the lower case items{i, j, k, l} arevectors (e.g. {}, respectively) and the upper case ones {I,J,K}={σ1,σ2,σ3} arebivectors (e.g., respectively) and theHodge star operator★ =i j k l is the pseudo-scalar element. 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★ =i j k l 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 thegamma matrices in the examples given above. The formula defining the fifth gamma matrix () shows that it is the★ of a four-dimensional geometric algebra of the gamma matrices.
A convenientmnemonic for remembering the products of unit octonions is given by the diagram, which represents the multiplication table of Cayley and Graves.[7][12]This diagram with seven points and seven lines (the circle through 1, 2, and 3 is considered a line) is called theFano plane. The lines are directional. The seven points correspond to the seven standard basis elements of (see definitionbelow). Each pair of distinct points lies on a unique line and each line runs through exactly three points.
Let(a,b,c) 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
together withcyclic permutations. These rules together with
completely defines the multiplicative structure of the octonions. Each of the seven lines generates asubalgebra of isomorphic to the quaternionsH.
Theconjugate of an octonion
is given by
Conjugation is aninvolution of and satisfies(xy)* =y*x* (note the change in order).
Thereal part ofx is given by
and theimaginary part (sometimes called thepure part) by
The set of all purely imaginary octonionsspans a 7 dimensionalsubspace of denoted
Conjugation of octonions satisfies the equation
The product of an octonion with its conjugate,x*x =xx* , is always a nonnegative real number:
Using this, the norm of an octonion is defined as
This norm agrees with the standard 8 dimensionalEuclidean norm onℝ8.
The existence of a norm on implies the existence ofinverses for every nonzero element of The inverse ofx ≠ 0 , which is the unique octonionx−1 satisfyingx x−1 =x−1x = 1 , is given by
Any octonionx can be decomposed into its real part and imaginary part:
also sometimes called scalar and vector parts.
We define theunit vectoru corresponding tox as
. It is a pure octonion of norm 1.
It can be proved[13] that any non-zero octonion can be written as:
thus providing a polar form.
Octonionic multiplication is neithercommutative:
norassociative:
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 isisomorphic toℝ,ℂ, orℍ, all of which are associative. Because of their non-associativity, octonions cannot be represented by a subalgebra of amatrix ring overℝ, unlike the real numbers, complex numbers, and quaternions.
The octonions do retain one important property shared byℝ,ℂ, andℍ: the norm on satisfies
This equation means that the octonions form acomposition algebra. The higher-dimensional algebras defined by the Cayley–Dickson construction (starting with thesedenions) all fail to satisfy this property. They all havezero 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 byHurwitz,ℝ,ℂ, orℍ, and are the only normed division algebras over the real numbers. These four algebras also form the only alternative, finite-dimensionaldivision algebras over the real numbers (up to an isomorphism).
Not being associative, the nonzero elements of do not form agroup. They do, however, form aloop, specifically aMoufang loop.
Thecommutator of two octonionsx andy is given by
This is antisymmetric and imaginary. If it is considered only as a product on the imaginary subspace it defines a product on that space, theseven-dimensional cross product, given by
Like thecross product in three dimensions this is a vector orthogonal tox andy with magnitude
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.[14]
Anautomorphism,A, of the octonions is an invertiblelinear transformation of which satisfies
The set of all automorphisms of forms a group calledG2 .[15] The groupG2 is asimply connected,compact, realLie group of dimension 14. This group is the smallest of the exceptional Lie groups and is isomorphic to thesubgroup ofSpin(7) that preserves any chosen particular vector in its 8 dimensional real spinor representation. The groupSpin(7) is in turn a subgroup of the group of isotopies described below.
See also:PSL(2,7) – theautomorphism group of the Fano plane.
Anisotopy of an algebra is a triple ofbijective linear mapsa,b,c such that ifxy =z thena(x)b(y) =c(z). Fora =b =c 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 groupSpin8(ℝ), witha,b,c acting as the three 8 dimensional representations.[16] The subgroup of elements wherec fixes the identity is the subgroupSpin7(ℝ), and the subgroup wherea,b,c all fix the identity is the automorphism groupG2 .
Just as quaternions can berepresented as matrices, octonions can be represented as tables of quaternions. Specifically, because any octonion can be defined a pair of quaternions, we represent the octonion as:
Using a slightly modified (non-associative) quaternionic matrix multiplication:we can translate octonion addition and multiplication to the respective operations on quaternionic matrices.[6]
The octonions play a significant role in the classification and construction of other mathematical entities. For example, theexceptional Lie groupG2 is the automorphism group of the octonions, and the other exceptional Lie groupsF4,E6,E7 andE8 can be understood as the isometries of certainprojective planes defined using the octonions.[17] The set ofself-adjoint 3 × 3 octonionicmatrices, equipped with a symmetrized matrix product, defines theAlbert algebra. Indiscrete mathematics, the octonions provide an elementary derivation of theLeech lattice, and thus they are closely related to thesporadic simple groups.[18][19]
Applications of the octonions to physics have largely been conjectural. For example, in the 1970s, attempts were made to understandquarks by way of an octonionicHilbert space.[20] It is known that the octonions, and the fact that only four normed division algebras can exist, relates to thespacetime dimensions in whichsupersymmetricquantum field theories can be constructed.[21][22] Also, attempts have been made to obtain theStandard Model of elementary particle physics from octonionic constructions, for example using the "Dixon algebra"[23][24]
Octonions have also arisen in the study ofblack hole entropy,quantum information science,[25][26]string theory,[27] andimage processing.[28]
Octonions have been used in solutions to thehand eye calibration problem inrobotics.[29]
Deep octonion networks provide a means of efficient and compact expression in machine learning applications.[30][31]
There are several natural ways to choose an integral form of the octonions. The simplest is just to take the octonions whose coordinates areintegers. This gives a nonassociative algebra over the integers called the Gravesian octonions. However it is not amaximal 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 byKirmse (1924), 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 extendedquadratic residue code of length 8 over the field of two elements, given by∅,(∞124) and its images under adding a constantmodulo 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 asCoxeter (1946) 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 theE8 lattice rescaled by a factor of1⁄√2. 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 octonionsa andb ≠ 0, we can findq andr witha =qb +r, where the remainderr has norm less than that ofb.
In the integral octonions, all leftideals and right ideals are 2-sided ideals, and the only 2-sided ideals are theprincipal idealsnO wheren 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 normmn can be written as a product of integral octonions of normsm andn.
The automorphism group of the integral octonions is the groupG2(F2) oforder 12,096, which has asimple subgroup ofindex 2 isomorphic to the unitary group2A2(32). The isotopy group of the integral octonions is the perfect double cover of the group of rotations of theE8 lattice.
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