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Octonion

Octonions
Symbol
TypeHypercomplex algebra
Unitse0, ..., e7
Multiplicative identitye0
Main properties
Common systems
Less common systems

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 O or blackboard bold . 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[clarification needed] 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 December 1843 by John T. Graves, inspired by his friend William 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 than Arthur Cayley's article.[3] The octonions were discovered independently by Cayley[4] and are sometimes referred to as Cayley numbers or the Cayley algebra. Hamilton described the early history of Graves's discovery.[5]

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:

where e0 is the scalar or real element; it may be identified with the real number 1 . That is, every octonion x can be written in the form

with real coefficients xi.

Cayley–Dickson construction

Main article: Cayley–Dickson construction

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]

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 (a, b) and (c, d) is defined by

where z* denotes the conjugate of the quaternion z. This definition is equivalent to the one given above when the eight unit octonions are identified with the pairs

(1, 0), (i, 0), (j, 0), (k, 0), (0, 1), (0, i), (0, j), (0, k)

Arithmetic and operations

Addition and subtraction

Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions.

Multiplication

Multiplication of octonions 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 (given both by Arthur Cayley in 1845 and John T. Graves in 1843):[7]

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 e0 is an operand.

The table can be summarized as follows:[8]

where δℓm is the Kronecker delta (equal to 1 if ℓ = m, and 0 for ℓ ≠ m), and εℓmn is a completely antisymmetric tensor with value +1 when ℓ 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 of permutations of the indices, but −1 for any odd permutations 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 with e0 = 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 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 (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) with e1e2 = e4 by using the triangular multiplication diagram, or Fano plane below that also shows the sorted list of 1 2 4 based 7-cycle triads and its associated multiplication matrices in both en and format.

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 e∞ = 1 and e1e2 = e4, and all expressions obtained from this by adding a constant (modulo 7) to all subscripts: In other words using the seven triples (1 2 4) (2 3 5) (3 4 6) (4 5 0) ( 5 6 1) (6 0 2) (0 1 3) . These are the nonzero codewords of the quadratic residue code of length 7 over the Galois field of two elements, GF(2). 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 .[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 field GF(7) of seven elements.

The Fano plane shown above with and IJKL multiplication matrices also includes the geometric algebra basis with signature (− − − −) and is given in terms of the following 7 quaternionic triples (omitting the scalar identity element):

(I , j , k ) , ( i , J , k) , ( i , j , K) , (I , J , K ) , (★I , i , l ) , (★J , j , l ), (★K , k , l)

or alternatively:

(★★★

in which the lower case items {i, j, k, l} are vectors (e.g. {}, respectively) and the upper case ones {I,J,K}={σ1,σ2,σ3} are bivectors (e.g. , respectively) and the Hodge 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 the gamma 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.

Fano plane mnemonic

A mnemonic for the products of the unit octonions[11]
A 3D mnemonic visualization showing the 7 triads as hyperplanes through the real (e0) vertex of the octonion example given above[11]

A convenient mnemonic 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 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 (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

ab = c and ba = −c

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 H.

Conjugate, norm, and inverse

The conjugate of an octonion

is given by

Conjugation is an involution of and satisfies (xy)* = y*x* (note the change in order).

The real part of x is given by

and the imaginary part by

The set of all purely imaginary octonions spans a 7 dimensional subspace 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 can be defined as

This norm agrees with the standard 8 dimensional Euclidean norm on ℝ8.

The existence of a norm on implies the existence of inverses for every nonzero element of The inverse of x ≠ 0 , which is the unique octonion x−1 satisfying x x−1 = x−1x = 1 , is given by

Properties

Octonionic multiplication is neither commutative:

ei ej = −ej ei ≠ ej ei if i, j are distinct and non-zero,

nor associative:

(ei ej) ek = −ei (ej ek) ≠ ei(ej ek) if i, j, k are distinct, non-zero and ei ej ≠ ±ek.

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 ℝ, ℂ, or ℍ, all of which are associative. Because of their non-associativity, octonions cannot be represented by a subalgebra of a matrix ring over ℝ, unlike the real numbers, complex numbers, and quaternions.

The octonions do retain one important property shared by ℝ, ℂ, and ℍ: the norm on satisfies

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, ℝ, ℂ, or ℍ, 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 an 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 x and y 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, the seven-dimensional cross product, given by

Like the cross product in three dimensions this is a vector orthogonal to x and y 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.[13]

Automorphisms

An automorphism, A, of the octonions is an invertible linear transformation of which satisfies

The set of all automorphisms of forms a group called G2 .[14] The group G2 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 Spin(7) that preserves any chosen particular vector in its 8 dimensional real spinor representation. The group Spin(7) is in turn a subgroup of the group of isotopies described below.

See also: PSL(2,7) – the automorphism group of the Fano plane.

Isotopies

An isotopy of an algebra is a triple of bijective linear maps a, b, c such that if xy = z then a(x)b(y) = c(z). For a = 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 group Spin8(ℝ), with a, b, c acting as the three 8 dimensional representations.[15] The subgroup of elements where c fixes the identity is the subgroup Spin7(ℝ), and the subgroup where a, b, c all fix the identity is the automorphism group G2 .

Applications

The octonions play a significant role in the classification and construction of other mathematical entities. For example, the exceptional Lie group G2 is the automorphism group of the octonions, and the other exceptional Lie groups F4, E6, E7 and E8 can be understood as the isometries of certain projective planes defined using the octonions.[16] 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.[17][18]

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.[19] 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.[20][21] Also, attempts have been made to obtain the Standard Model of elementary particle physics from octonionic constructions, for example using the "Dixon algebra" [22][23]

Octonions have also arisen in the study of black hole entropy, quantum information science,[24][25] string theory,[26] and image processing.[27]

Octonions have been used in solutions to the hand eye calibration problem in robotics.[28]

Deep octonion networks provide a means of efficient and compact expression in machine learning applications.[29][30]

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 Kirmse (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

∅ (∞124) (∞235) (∞346) (∞450) (∞561) (∞602) (∞013) (∞0123456) (0356) (1460) (2501) (3612) (4023) (5134) (6245)

of the extended quadratic residue code of length 8 over the field of two elements, given by ∅, (∞124) 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 Coxeter (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 the E8 lattice rescaled by a factor of 1⁄√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 octonions a and b ≠ 0, we can find q and r with a = qb + r, where the remainder r has norm less than that of b.

In the integral octonions, all left ideals and right ideals are 2-sided ideals, and the only 2-sided ideals are the principal ideals nO where n 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 mn can be written as a product of integral octonions of norms m and n.

The automorphism group of the integral octonions is the group G2(F2) of order 12,096, which has a simple subgroup of index 2 isomorphic to the unitary group 2A2(32). The isotopy group of the integral octonions is the perfect double cover of the group of rotations of the E8 lattice.

See also

Notes

  1. ^ (Baez 2002, p. 1)
  2. ^ Sabadini, Irene; Shapiro, Michael; Sommen, Franciscus (2009-04-21), Hypercomplex Analysis, Springer Science & Business Media, ISBN 978-3-7643-9893-4
  3. ^ (Graves 1845)
  4. ^ Cayley, Arthur (1845), "On Jacobi's Elliptic functions, in reply to the Rev. Brice Bronwin; and on Quaternions", Philosophical Magazine, 26 (172): 208–211, doi:10.1080/14786444508645107. Appendix reprinted in The Collected Mathematical Papers, Johnson Reprint Co., New York, 1963, p. 127
  5. ^ Hamilton (1848), "Note, by Sir W. R. Hamilton, respecting the researches of John T. Graves, Esq.", Transactions of the Royal Irish Academy, 21: 338–341
  6. ^ "Ensembles de nombre" (PDF) (in French), Forum Futura-Science, 6 September 2011, retrieved 11 October 2024
  7. ^ a b Gentili, G.; Stoppato, C.; Struppa, D.C.; Vlacci, F. (2009), "Recent developments for regular functions of a hypercomplex variable", in Sabadini, I.; Shapiro, M.; Sommen, F. (eds.), Hypercomplex Analysis, Birkhäuser, p. 168, ISBN 978-3-7643-9892-7 – via Google books
  8. ^ Sabinin, L.V.; Sbitneva, L.; Shestakov, I.P. (2006), "§17.2 Octonion algebra and its regular bimodule representation", Non-Associative Algebra and its Applications, Boca Raton, FL: CRC Press, p. 235, ISBN 0-8247-2669-3 – via Google books
  9. ^ Abłamowicz, Rafał; Lounesto, Pertti; Parra, Josep M. (1996), "§ Four ocotonionic basis numberings", Clifford Algebras with Numeric and Symbolic Computations, Birkhäuser, p. 202, ISBN 0-8176-3907-1 – via Google books
  10. ^ Schray, Jörg; Manogue, Corinne A. (January 1996), "Octonionic representations of Clifford algebras and triality", Foundations of Physics, 26 (1): 17–70, arXiv:hep-th/9407179, Bibcode:1996FoPh...26...17S, doi:10.1007/BF02058887, S2CID 119604596
    Available as Schray, Jörg; Manogue, Corinne A. (1996), "Octonionic representations of Clifford algebras and triality", Foundations of Physics, 26 (1): 17–70, arXiv:hep-th/9407179, Bibcode:1996FoPh...26...17S, doi:10.1007/BF02058887, in particular Figure 1 (.png), arXiv (image)
  11. ^ a b (Baez 2002, p. 6)
  12. ^ Dray, Tevian & Manogue, Corinne A. (2004), "Chapter 29: Using octonions to describe fundamental particles", in Abłamowicz, Rafał (ed.), Clifford Algebras: Applications to mathematics, physics, and engineering, Birkhäuser, Figure 29.1: Representation of multiplication table on projective plane. p. 452, ISBN 0-8176-3525-4 – via Google books
  13. ^ Baez (2002), pp. 37–38
  14. ^ (Conway & Smith 2003, ch 8.6)
  15. ^ (Conway & Smith 2003, ch 8)
  16. ^ Baez (2002), section 4.
  17. ^ Wilson, Robert A. (2009-09-15), "Octonions and the Leech lattice" (PDF), Journal of Algebra, 322 (6): 2186–2190, doi:10.1016/j.jalgebra.2009.03.021
  18. ^ Wilson, Robert A. (2010-08-13), "Conway's group and octonions" (PDF), Journal of Group Theory, 14: 1–8, doi:10.1515/jgt.2010.038, S2CID 16590883
  19. ^ Günaydin, M.; Gürsey, F. (1973), "Quark structure and octonions", Journal of Mathematical Physics, 14 (11): 1651–1667, Bibcode:1973JMP....14.1651G, doi:10.1063/1.1666240
    Günaydin, M.; Gürsey, F. (1974), "Quark statistics and octonions", Physical Review D, 9 (12): 3387–3391, Bibcode:1974PhRvD...9.3387G, doi:10.1103/PhysRevD.9.3387
  20. ^ Kugo, Taichiro; Townsend, Paul (1983-07-11), "Supersymmetry and the division algebras", Nuclear Physics B, 221 (2): 357–380, Bibcode:1983NuPhB.221..357K, doi:10.1016/0550-3213(83)90584-9
  21. ^ Baez, John C.; Huerta, John (2010), "Division Algebras and Supersymmetry I", in Doran, R.; Friedman, G.; Rosenberg, J. (eds.), Superstrings, Geometry, Topology, and C*-algebras, American Mathematical Society, arXiv:0909.0551
  22. ^ Wolchover, Natalie (2018-07-20), "The peculiar math that could underlie the laws of nature", Quanta Magazine, retrieved 2018-10-30
  23. ^ Furey, Cohl (2012-07-20), "Unified theory of ideals", Physical Review D, 86 (2): 025024, arXiv:1002.1497, Bibcode:2012PhRvD..86b5024F, doi:10.1103/PhysRevD.86.025024, S2CID 118458623
    Furey, Cohl (2018-10-10), "Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra", Physics Letters B, 785: 84–89, arXiv:1910.08395, Bibcode:2018PhLB..785...84F, doi:10.1016/j.physletb.2018.08.032, S2CID 126205768
    Stoica, O.C. (2018), "Leptons, quarks, and gauge from the complex Clifford algebra ", Advances in Applied Clifford Algebras, 28: 52, arXiv:1702.04336, doi:10.1007/s00006-018-0869-4, S2CID 125913482
    Gresnigt, Niels G. (2017-11-21), Quantum groups and braid groups as fundamental symmetries, European Physical Society conference on High Energy Physics, 5–12 July 2017, Venice, Italy, arXiv:1711.09011
    Dixon, Geoffrey M. (1994), Division Algebras: Octonions, quaternions, complex numbers, and the algebraic design of physics, Springer-Verlag, doi:10.1007/978-1-4757-2315-1, ISBN 978-0-7923-2890-2, OCLC 30399883
    Baez, John C. (2011-01-29), "The Three-Fold Way (part 4)", The n-Category Café, retrieved 2018-11-02
  24. ^ Borsten, Leron; Dahanayake, Duminda; Duff, Michael J.; Ebrahim, Hajar; Rubens, Williams (2009), "Black holes, qubits and octonions", Physics Reports, 471 (3–4): 113–219, arXiv:0809.4685, Bibcode:2009PhR...471..113B, doi:10.1016/j.physrep.2008.11.002, S2CID 118488578
  25. ^ Stacey, Blake C. (2017), "Sporadic SICs and the Normed Division Algebras", Foundations of Physics, 47 (8): 1060–1064, arXiv:1605.01426, Bibcode:2017FoPh...47.1060S, doi:10.1007/s10701-017-0087-2, S2CID 118438232
  26. ^ "Beyond space and time: 8D – Surfer's paradise", New Scientist
  27. ^ Jacome, Roman; Mishra, Kumar Vijay; Sadler, Brian M.; Arguello, Henry (2024), "Octonion Phase Retrieval", IEEE Signal Processing Letters, 31: 1615, arXiv:2308.15784, Bibcode:2024ISPL...31.1615J, doi:10.1109/LSP.2024.3411934
  28. ^ Wu, J.; Sun, Y.; Wang and, M.; Liu, M. (June 2020), "Hand-Eye Calibration: 4-D Procrustes Analysis Approach", IEEE Transactions on Instrumentation and Measurement, 69 (6): 2966–81, Bibcode:2020ITIM...69.2966W, doi:10.1109/TIM.2019.2930710, S2CID 201245901
  29. ^ Wu, J.; Xu, L.; Wu, F.; Kong, Y.; Senhadji, L.; Shu, H. (2020), "Deep octonion networks", Neurocomputing, 397: 179–191, doi:10.1016/j.neucom.2020.02.053, S2CID 84186686, hal-02865295
  30. ^ Bojesomo, Alabi; Liatsis, Panos; Almarzouqi, Hasan (2023), "Marine Debris Segmentation Using a Parameter Efficient Octonion-Based Architecture", IEEE Geoscience and Remote Sensing Letters, 20: 1–5, Bibcode:2023IGRSL..2021177B, doi:10.1109/lgrs.2023.3321177

References

Quelle: http://en.wikipedia.org/wiki/Octonions