In mathematics, the octonions are a nonassociative extension of the quaternions. They form an 8-dimensional normed division algebra over the real numbers. The octonion algebra is often denoted O. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || . || satisfying ||xy|| = ||x|| ||y|| for all x and y in A. While the definition allows normed division algebras to be infinite-dimensional, this, in... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

Lacking the desirable property of associativity, the octonions receive far less attention than the quaternions. Despite this, the octonions retain importance for being related to a number of exceptional structures in mathematics, among them the exceptional Lie groups. In mathematics, a simple Lie group is a Lie group which is also a simple group. ...

History

The octonions were discovered in 1843 by John T. Graves, a friend of William Hamilton, who called them octaves. They were discovered independently by Arthur Cayley, who published the first paper on them in 1845. They are sometimes referred to as Cayley numbers or the Cayley algebra. 1843 was a common year starting on Sunday (see link for calendar). ... William Rowan Hamilton Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer. ... Arthur Cayley (August 16, 1821 - January 26, 1895) was a British mathematician. ... 1845 was a common year starting on Wednesday (see link for calendar). ...

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 {1, i, j, k, l, li, lj, lk}. That is, every octonion x can be written in the form In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

x = x₀ + x₁ i + x₂ j + x₃ k + x₄ l + x₅ li + x₆ lj + x₇ lk.

with real coefficients x_a.

Addition of octonions is accomplished by adding corresponding coefficients, as with the complex numbers and quaternions. By linearity, multiplication of octonions is completely determined by the multiplication table for the unit octonions given below. In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit , satisfying . ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, a multiplication table is used to define a multiplication operation for an algebraic system. ...

(Note that the basis for the octonions given here is not nearly as universal as the standard basis for the quaternions, however, nearly all other choices differ from this one only in order and sign.) In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

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 (a, b) and (c, d) is defined by In mathematics, the Cayley-Dickson construction produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. ...

(a, b)(c, d) = (ac − db^*, a^*d + cb)

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)

Fano plane mnemonic

A convenient mnemonic for remembering the products of unit octonions is given by the following diagram: A mnemonic (Pronounced in American English, in British English) is a memory aid. ...

This diagram with seven points and seven lines (the circle through i, j, and k is considered a line) is called the Fano plane. Note that the lines are oriented in this diagram. The seven points correspond to the seven standard basis elements of Im(O). Note that each pair of distinct points lies on a unique line and each line runs through exactly three points. A finite geometry is any geometric system that has only a finite number of 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 A cyclic permutation is a permutation that shifts all elements of given ordered set by a fixed offset, with the elements shifted off the end inserted back at the beginning in the same order, i. ...

• 1 is the multiplicative identity,
• e² = −1 for each point in the diagram

completely defines the algebraic structure of the octonions. Note that each of the seven lines generates a subalgebra of O isomorphic to the quaternions H.

Conjugate, norm, and inverse

The conjugate of an octonion

x = x₀ + x₁ i + x₂ j + x₃ k + x₄ l + x₅ li + x₆ lj + x₇ lk

is given by

x^* = x₀ − x₁ i − x₂ j − x₃ k − x₄ l − x₅ li − x₆ lj − x₇ lk.

Conjugation is an involution of O and satisfies (xy)^* = y^*x^* (note the change in order). In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ...

The real part of x is defined as ½(x + x^*) = x₀ and the imaginary part as ½(x - x^*). The set of all purely imaginary octonions span a 7 dimension subspace of O, denoted Im(O).

The norm of the octonion x is defined as

The square root is well-defined here as x^*x = xx^* is always a nonnegative real number: In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is . ...

Note that this norm agrees with the standard Euclidean norm on R⁸. In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

The existence of a norm on O implies the existence of inverses for every nonzero element of O. The inverse of x ≠ 0 is given by In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ...

It satisfies xx⁻¹ = x⁻¹x = 1.

Properties

Octonionic multiplication is neither commutative: In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

ij = − ji

nor associative: In mathematics, associativity is a property that a binary operation can have. ...

(ij)l = − i(jl)

They 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 O is isomorphic to R, C, or H, all of which are associative. In abstract algebra, an algebra is called alternative if (xx)y=x(xy) and y(xx)=(yx)x for all x and y in the algebra, that is, if the multiplication is alternative. ... In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to A. Since the axioms of algebraic structures in universal algebra are described by equational laws, the... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

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

This implies that the octonions form a nonassociative normed division algebra. The higher-dimensional algebras defined by the Cayley-Dickson construction (e.g. the sedenions) all fail to satisfy this property. They all have zero divisors. In mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || . || satisfying ||xy|| = ||x|| ||y|| for all x and y in A. While the definition allows normed division algebras to be infinite-dimensional, this, in... In mathematics, the Cayley-Dickson construction produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. ... The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ... In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ...

It turns out that the only normed division algebras over the reals are R, C, H, and O. These four algebras also form the only alternative, finite-dimensional division algebra over the reals (up to isomorphism). In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. ... In mathematics, the term up to xxxx is used to describe a situation in which members of an equivalence class can be regarded as a single entity for some purpose. ...

Not being associative, the nonzero elements of O do not form a group. They do, however, form a quasigroup, indeed a Moufang loop. In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ...

Automorphisms

An automorphism, A, of the octonions is an invertible linear transformation of O which satisfies In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

A(xy) = A(x)A(y).

The set of all automorphisms of O forms a group called G₂. The group G₂ is a simply connected, compact, real Lie group of dimension 14. This group is the smallest of the five exceptional Lie groups. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, G2 is the name of a Lie group and also its Lie algebra . ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... Several specialized usages of the terms compact and compactness exist. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, a simple Lie group is a Lie group which is also a simple group. ...

See also: PSL(2,7) - the automorphism group of the Fano plane. The projective special linear group G = PSL(2,7) is a finite group in mathematics that has important applications in algebra, geometry, and number theory. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

Related topics

• hypercomplex numbers
  • quaternions
  • sedenions
• triality
• Spin(8)

In mathematics, hypercomplex numbers are extensions of the complex numbers constructed by means of abstract algebra, such as quaternions, tessarines, coquaternions, octonions, biquaternions and sedenions. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ... Categories: Stub | Lie groups ... In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. ...

References

• John Baez, The Octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205 (http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/home.html). Online HTML version at http://math.ucr.edu/home/baez/octonions/.
• Jonh Conway and Derek Smith, On Octonions and Quaternions, A K Peters, Natick, MA (2003). ISBN 1-56881-134-9.