Octonion

In mathematics, the octonions are a nonassociative extension of the quaternions. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, associativity is a property that a Binary operation can have Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Their 8-dimensional normed division algebra over the real numbers is the widest possible that can be obtained from the Cayley-Dickson construction. In Mathematics, a normed division algebra A is a Division algebra over the real or complex numbers which is also a Normed vector In Mathematics, the real numbers may be described informally in several different ways The octonion algebra is often denoted O, or in blackboard bold by $\mathbb{O}$ . Blackboard bold is a Typeface style often used for certain symbols in Mathematics and Physics texts in which certain lines of the symbol (usually vertical

Possibly because they don't offer an associative multiplication, the octonions receive at times less attention than the quaternions. Despite this lack of popularity, they are related to a number of exceptional structures in mathematics, among them the exceptional Lie groups. In Mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected Normal subgroups Additionally, octonions have applications in fields such as string theory, special relativity, and quantum logic. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Mathematical physics and Quantum mechanics, quantum logic is a set of rules for Reasoning about propositions which takes the principles of

1 History
2 Definition
3 Properties
- 3.1 Automorphisms
4 Applications in physics
5 Quotes
6 See also
7 References

History

The octonions were discovered in 1843 by John T. Graves, a friend of William Hamilton, who called them octaves. Year 1843 ( MDCCCXLIII) was a Common year starting on Sunday (link will display the full calendar of the Gregorian Calendar (or a Common John Thomas Graves Esq (1806 - 1870 was an Irish Jurist and Mathematician. Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who They were discovered independently by Arthur Cayley, who published the first paper on them in 1845. Arthur Cayley ( August 16 1821 - January 26 1895) was a British Mathematician. Year 1845 ( MDCCCXLV) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Common They are sometimes referred to as Cayley numbers or the Cayley algebra.

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, il, jl, kl}. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics That is, every octonion x can be written in the form

$x = x_0 + x_1\,i + x_2\,j + x_3\,k + x_4\,l + x_5\,il + x_6\,jl + x_7\,kl.$

with real coefficients x_a.

Addition of octonions is accomplished by adding corresponding coefficients, as with the complex numbers and quaternions. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician By linearity, multiplication of octonions is completely determined by the multiplication table for the unit octonions given below.

1	i	j	k	l	il	jl	kl
i	−1	k	−j	il	−l	−kl	jl
j	−k	−1	i	jl	kl	−l	−il
k	j	−i	−1	kl	−jl	il	−l
l	−il	−jl	−kl	−1	i	j	k
il	l	−kl	jl	−i	−1	−k	j
jl	kl	l	−il	−j	k	−1	−i
kl	−jl	il	l	−k	−j	i	−1

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. Basis vector redirects here For basis vector in the context of crystals see Crystal structure.

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

(a, b)(c, d) = (a c - d * b, d a + b c *)

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 simple mnemonic for the products of the unit octonions.

A convenient mnemonic for remembering the products of unit octonions is given by the following diagram at the right. A mnemonic device (nəˈmɒnɪk is a Memory aid Commonly met mnemonics are often verbal something such as a very short poem or a special word used to help a person remember This diagram with seven points and seven lines (the circle through i, j, and k is considered a line) is called the Fano plane. In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each The lines are oriented in this diagram. The seven points correspond to the seven standard basis elements of Im(O). 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. A cyclic Permutation is built from one or more sets of elements in Cyclic order. These rules together with

1 is the multiplicative identity,
$e 2 = - 1$ for each point in the diagram

completely defines the multiplicative structure of the octonions. Each of the seven lines generates a subalgebra of O isomorphic to the quaternions H.

Conjugate, norm, and inverse

The conjugate of an octonion

$x = x_0 + x_1\,i + x_2\,j + x_3\,k + x_4\,l + x_5\,il + x_6\,jl + x_7\,kl$

is given by

$x^* = x_0 - x_1\,i - x_2\,j - x_3\,k - x_4\,l - x_5\,il - x_6\,jl - x_7\,kl.$

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

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

$\|x\| = \sqrt{x^{*} x}$

The square root is well-defined here as $x * x = x x *$ is always a nonnegative real number:

$\|x\|^2 = x^{*}x = x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2$

This norm agrees with the standard Euclidean norm on R⁸. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length

The existence of a norm on O implies the existence of inverses for every nonzero element of O. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to The inverse of x ≠ 0 is given by

$x^{-1} = \frac{x^{*}}{\|x\|^2}$

It satisfies $x x - 1 = x - 1 x = 1$ .

Properties

Octonionic multiplication is neither commutative:

$ij = -ji \neq ji\,$

nor associative:

$(ij)l = -i(jl) \neq i(jl)\,$

The octonions do satisfy a weaker form of associativity: they are alternative. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, associativity is a property that a Binary operation can have In Abstract algebra, an alternative algebra is an algebra in which multiplication need not be Associative, only alternative. This means that the subalgebra generated by any two elements is associative. In Algebra (mathematics, the word "algebra" usually means a Vector space or module equipped with an additional bilinear operation In Mathematics, associativity is a property that a Binary operation can have 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 isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

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

$\|xy\| = \|x\|\|y\|$

This implies that the octonions form a nonassociative normed division algebra. In Mathematics, a normed division algebra A is a Division algebra over the real or complex numbers which is also a Normed vector The higher-dimensional algebras defined by the Cayley-Dickson construction (e. g. the sedenions) all fail to satisfy this property. In Abstract algebra, sedenions form a 16- dimensional algebra over the reals. They all have zero divisors. In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0

Wider number systems exist which have a multiplicative modulus (e. g. 16 dimensional conic sedenions from the Musean hypernumbers program). In Abstract algebra, sedenions form a 16- dimensional algebra over the reals. Musean hypernumbers are an algebraic concept envisioned by Charles A Their modulus is defined differently from their norm, and they also contain zero divisors. In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero 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 algebras 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 phrase " up to xxxx" indicates that members of an Equivalence class are to 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 loop, indeed a Moufang loop. In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Mathematics, a Moufang loop is a special kind of Algebraic structure.

Automorphisms

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

$A(xy) = A(x)A(y).\,$

The set of all automorphisms of O forms a group called G₂. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element The group G₂ is a simply connected, compact, real Lie group of dimension 14. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group This group is the smallest of the exceptional Lie groups. In Mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected Normal subgroups

See also: PSL(2,7) - the automorphism group of the Fano plane. In Mathematics, the Projective special linear group PSL(27 is a finite Simple group that has important applications in Algebra, In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

Applications in physics

Octonions are used for a nonassociative generalization of quantum mechanics. For example, the Heisenberg equation of motion is modified using a nonassociative commutator (see references below).

Quotes

The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative. — John Baez

References

John Baez, The Octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205. John Carlos Baez (born 1961 is an American mathematical physicist at the University of California Riverside. In Mathematics, the split-octonions are a Nonassociative extension of the Quaternions (or the Split-quaternions. In Mathematics, an octonion algebra over a field F is an Algebraic structure which is an 8- dimensional Composition algebra The term hypercomplex number has been used in Mathematics for the elements of algebras that extend or go beyond Complex number arithmetic Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Abstract algebra, sedenions form a 16- dimensional algebra over the reals. Musean hypernumbers are an algebraic concept envisioned by Charles A In Mathematics, triality is a relationship between three Vector spaces analogous to the duality relation between Dual vector spaces Most commonly In Mathematics, SO(8 is the Special orthogonal group acting on eight-dimensional Euclidean space. In Mathematics, the seven-dimensional cross product is a Binary operation on vectors in a seven-dimensional Euclidean space. John Carlos Baez (born 1961 is an American mathematical physicist at the University of California Riverside. Online HTML version at http://math.ucr.edu/home/baez/octonions/.
Conway, John Horton, and Smith, Derek A. John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups , (2003) On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A. K. Peters, Ltd. ISBN 1-56881-134-9. (Review).

For physics on octonion arithmetic see e. g.

V. Dzhunushaliev, Toy Models of a Nonassociative Quantum Mechanics, Advances in High Energy Physics, vol. 2007, Article ID 12387, 10 pages, 2007. doi:10. 1155/2007/12387; arXiv:0706. 2398 [quant-ph].

Dictionary

-noun

(mathematics) A nonassociative extension of the quaternions

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