OctonionFrom Wikipedia, the free encyclopedia

Contents

1 Algebra over a eld 11.1 Denition and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 First example: The complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 A motivating example: quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.4 Another motivating example: the cross product . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Algebra homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Subalgebras and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.3 Extension of scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Kinds of algebras and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.1 Unital algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Zero algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.3 Associative algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.4 Non-associative algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Algebras and rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Structure coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Classication of low-dimensional algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Alternative algebra 82.1 The associator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Associative property 113.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

i

ii CONTENTS

3.2 Generalized associative law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4.1 Rule of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.2 Truth functional connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5 Non-associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5.1 Nonassociativity of oating point calculation . . . . . . . . . . . . . . . . . . . . . . . . . 163.5.2 Notation for non-associative operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Flexible algebra 194.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Non-associative algebra 205.1 Algebras satisfying identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.1.1 Associator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3 Free non-associative algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.4 Associated algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.4.1 Derivation algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.4.2 Enveloping algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Octonion 256.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.1.1 CayleyDickson construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.1.2 Fano plane mnemonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.1.3 Conjugate, norm, and inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2.1 Commutator and cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2.2 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2.3 Isotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.3 Integral octonions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

CONTENTS iii

6.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 326.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Chapter 1

Algebra over a eld

This article is about vector spaces equipped with some kind of multiplication. For other uses of the term algebra,see algebra.

In mathematics, an algebra over a eld is a vector space equipped with a bilinear product. Thus, an algebra overa eld is a set, together with operations of multiplication, addition, and scalar multiplication, by elements of theunderlying eld, that satisfy the axioms of implied by vector space and bilinear.[1]

An algebra such that the product is associative and has an identity is therefore a ring that is also a vector space, andthus equipped with a eld of scalars. Such an algebra is called here a unital associative algebra for clarity, becausethere are nonunital and nonassociative algebras.One may generalize this notion by replacing the eld of scalars by a commutative ring, and thus dening an algebraover a ring.Because of the ubiquity of associative algebras, and because many textbooks teach more associative algebra thannonassociative algebra, it is common for authors to use the term algebra to mean associative algebra. However, thisdoes not diminish the importance of nonassociative algebras, and there are texts that give both structures and namesequal priority.

1.1 Denition and motivation

1.1.1 First example: The complex numbers

Any complex number may be written a + bi, where a and b are real numbers and i is the imaginary unit. In otherwords, a complex number is represented by the vector (a, b) over the eld of real numbers. So the complex numbersform a two-dimensional real vector space, where addition is given by (a, b) + (c, d) = (a + c, b + d) and scalarmultiplication is given by c(a, b) = (ca, cb), where all of a, b, c and d are real numbers. We use the symbol tomultiply two vectors together, which we use complex multiplication to dene: (a, b) (c, d) = (ac bd, ad + bc).The following statements are basic properties of the complex numbers. Let x, y, z be complex numbers, and let a, bbe real numbers.

(x + y) z = (x z) + (y z). In other words, multiplying a complex number by the sum of twoother complex numbers, is the same as multiplying by each number in the sum, and then adding.

(ax) (by) = (ab) (x y). This shows that complex multiplication is compatible with the scalarmultiplication by the real numbers.

This example ts into the following denition by taking the eld K to be the real numbers, and the vector space A tobe the complex numbers.

1

2 CHAPTER 1. ALGEBRA OVER A FIELD

1.1.2 Denition

Let K be a eld, and let A be a vector space over K equipped with an additional binary operation from A A to A,denoted here by (i.e. if x and y are any two elements of A, x y is the product of x and y). Then A is an algebraover K if the following identities hold for any three elements x, y, and z of A, and all elements ("scalars") a and b ofK:

Right distributivity: (x + y) z = x z + y z

Left distributivity: x (y + z) = x y + x z

Compatibility with scalars: (ax) (by) = (ab) (x y).

These three axioms are another way of saying that the binary operation is bilinear. An algebra over K is sometimesalso called a K-algebra, and K is called the base eld ofA. The binary operation is often referred to asmultiplication inA. The convention adopted in this article is that multiplication of elements of an algebra is not necessarily associative,although some authors use the term algebra to refer to an associative algebra.Notice that when a binary operation on a vector space is commutative, as in the above example of the complexnumbers, it is left distributive exactly when it is right distributive. But in general, for non-commutative operations(such as the next example of the quaternions), they are not equivalent, and therefore require separate axioms.

1.1.3 A motivating example: quaternions

Main article: Quaternion

The real numbers may be viewed as a one-dimensional vector space with a compatible multiplication, and hence aone-dimensional algebra over itself. Likewise, as we saw above, the complex numbers form a two-dimensional vectorspace over the eld of real numbers, and hence form a two dimensional algebra over the reals. In both these examples,every non-zero vector has an inverse, making them both division algebras. Although there are no division algebras in 3dimensions, in 1843, the quaternions were dened and provided the now famous 4-dimensional example of an algebraover the real numbers, where one can not only multiply vectors, but also divide. Any quaternion may be written as(a, b, c, d) = a + bi + cj + dk. Unlike the complex numbers, the quaternions are an example of a non-commutativealgebra: for instance, (0,1,0,0) (0,0,1,0) = (0,0,0,1) but (0,0,1,0) (0,1,0,0) = (0,0,0,1).The quaternions were soon followed by several other hypercomplex number systems, which were the early examplesof algebras over a eld.

1.1.4 Another motivating example: the cross product

Main article: Cross product

Previous examples are associative algebras. An example of a nonassociative algebra is a three dimensional vectorspace equipped with the cross product. This is a simple example of a class of nonassociative algebras, which iswidely used in mathematics and physics, the Lie algebras.

1.2 Basic concepts

1.2.1 Algebra homomorphisms

Main article: Algebra homomorphism

Given K-algebras A and B, a K-algebra homomorphism is a K-linear map f: A B such that f(xy) = f(x) f(y) forall x,y in A. The space of all K-algebra homomorphisms between A and B is frequently written as

1.3. KINDS OF ALGEBRAS AND EXAMPLES 3

HomK-alg(A;B):

A K-algebra isomorphism is a bijective K-algebra morphism. For all practical purposes, isomorphic algebras dieronly by notation.

1.2.2 Subalgebras and ideals

Main article: Substructure

A subalgebra of an algebra over a eld K is a linear subspace that has the property that the product of any two of itselements is again in the subspace. In other words, a subalgebra of an algebra is a subset of elements that is closedunder addition, multiplication, and scalar multiplication. In symbols, we say that a subset L of a K-algebra A is asubalgebra if for every x, y in L and c in K, we have that x y, x + y, and cx are all in L.In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra.A left ideal of a K-algebra is a linear subspace that has the property that any element of the subspace multiplied onthe left by any element of the algebra produces an element of the subspace. In symbols, we say that a subset L of aK-algebra A is a left ideal if for every x and y in L, z in A and c in K, we have the following three statements.

1) x + y is in L (L is closed under addition),

2) cx is in L (L is closed under scalar multiplication),

3) z x is in L (L is closed under left multiplication by arbitrary elements).

If (3) were replaced with x z is in L, then this would dene a right ideal. A two-sided ideal is a subset that is botha left and a right ideal. The term ideal on its own is usually taken to mean a two-sided ideal. Of course when thealgebra is commutative, then all of these notions of ideal are equivalent. Notice that conditions (1) and (2) together areequivalent to L being a linear subspace of A. It follows from condition (3) that every left or right ideal is a subalgebra.It is important to notice that this denition is dierent from the denition of an ideal of a ring, in that here we requirethe condition (2). Of course if the algebra is unital, then condition (3) implies condition (2).

1.2.3 Extension of scalars

Main article: Extension of scalars

If we have a eld extension F/K, which is to say a bigger eld F that contains K, then there is a natural way to constructan algebra over F from any algebra over K. It is the same construction one uses to make a vector space over a biggereld, namely the tensor product VF := V K F . So if A is an algebra over K, then AF is an algebra over F.

1.3 Kinds of algebras and examplesAlgebras over elds come in many dierent types. These types are specied by insisting on some further axioms,such as commutativity or associativity of the multiplication operation, which are not required in the broad denitionof an algebra. The theories corresponding to the dierent types of algebras are often very dierent.

1.3.1 Unital algebra

An algebra is unital or unitary if it has a unit or identity element I with Ix = x = xI for all x in the algebra.

4 CHAPTER 1. ALGEBRA OVER A FIELD

1.3.2 Zero algebraAn algebra is called zero algebra if uv = 0 for all u, v in the algebra,[2] not to be confused with the algebra with oneelement. It is inherently non-unital (except in the case of only one element), associative and commutative.One may dene a unital zero algebra by taking the direct sum of modules of a eld (or more generally a ring) Kand a K-vector space (or module) V, and dening the product of every pair of elements of V to be zero. That is, if, k and u, v V, then ( + u) ( + v) = + (v + u). If e1, ... ed is a basis of V, the unital zero algebra is thequotient of the polynomial ring K[E1, ..., En] by the ideal generated by the EiEj for every pair (i, j).An example of unital zero algebra is the algebra of dual numbers, the unital zero R-algebra built from a one dimen-sional real vector space.These unital zero algebras may be more generally useful, as they allow to translate any general property of the algebrasto properties of vector spaces or modules. For example, the theory of Grbner bases was introduced by BrunoBuchberger for ideals in a polynomial ring R = K[x1, ..., xn] over a eld. The construction of the unital zero algebraover a free R-module allows to extend directly this theory as a Grbner basis theory for sub modules of a free module.This extension allows, for computing a Grbner basis of a submodule, to use, without any modication, any algorithmand any software for computing Grbner bases of ideals.

1.3.3 Associative algebraMain article: Associative algebra

the algebra of all n-by-n matrices over the eld (or commutative ring) K. Here the multiplication is ordinarymatrix multiplication.

Group algebras, where a group serves as a basis of the vector space and algebra multiplication extends groupmultiplication.

the commutative algebra K[x] of all polynomials over K. algebras of functions, such as theR-algebra of all real-valued continuous functions dened on the interval [0,1],or the C-algebra of all holomorphic functions dened on some xed open set in the complex plane. These arealso commutative.

Incidence algebras are built on certain partially ordered sets. algebras of linear operators, for example on a Hilbert space. Here the algebra multiplication is given by thecomposition of operators. These algebras also carry a topology; many of them are dened on an underlyingBanach space, which turns them into Banach algebras. If an involution is given as well, we obtain B*-algebrasand C*-algebras. These are studied in functional analysis.

1.3.4 Non-associative algebraMain article: Non-associative algebra

A non-associative algebra[3] (or distributive algebra) over a eld K is a K-vector space A equipped with a K-bilinearmap A A ! A . The usage of non-associative here is meant to convey that associativity is not assumed, but itdoes not mean it is prohibited. That is, it means not necessarily associative just as noncommutative means notnecessarily commutative.Examples detailed in the main article include:

Octonions Lie algebras Jordan algebras

1.4. ALGEBRAS AND RINGS 5

Alternative algebras Flexible algebras Power-associative algebras

1.4 Algebras and ringsThe denition of an associative K-algebra with unit is also frequently given in an alternative way. In this case, analgebra over a eld K is a ring A together with a ring homomorphism

: K ! Z(A);

where Z(A) is the center of A. Since is a ring morphism, then one must have either that A is the zero ring, or that is injective. This denition is equivalent to that above, with scalar multiplication

K A! A

given by

(k; a) 7! (k)a:

Given two such associative unital K-algebras A and B, a unital K-algebra morphism f: A B is a ring morphism thatcommutes with the scalar multiplication dened by , which one may write as

f(ka) = kf(a)

for all k 2 K and a 2 A . In other words, the following diagram commutes:

KA . B &

Af! B

1.5 Structure coecientsFor algebras over a eld, the bilinear multiplication from A A to A is completely determined by the multiplicationof basis elements of A. Conversely, once a basis for A has been chosen, the products of basis elements can be setarbitrarily, and then extended in a unique way to a bilinear operator on A, i.e., so the resulting multiplication satisesthe algebra laws.Thus, given the eld K, any nite-dimensional algebra can be specied up to isomorphism by giving its dimension(say n), and specifying n3 structure coecients ci,j,k, which are scalars. These structure coecients determine themultiplication in A via the following rule:

eiej =nX

k=1

ci;j;kek

where e1,...,en form a basis of A.Note however that several dierent sets of structure coecients can give rise to isomorphic algebras.

6 CHAPTER 1. ALGEBRA OVER A FIELD

When the algebra can be endowed with a metric, then the structure coecients are written with upper and lowerindices, so as to distinguish their transformation properties under coordinate transformations. Specically, lowerindices are covariant indices, and transform via pullbacks, while upper indices are contravariant, transforming underpushforwards. Thus, in mathematical physics, the structure coecients are often written ci,jk, and their dening ruleis written using the Einstein notation as

eiej = ci,jkek.

If you apply this to vectors written in index notation, then this becomes

(xy)k = ci,jkxiyj .

IfK is only a commutative ring and not a eld, then the same process works ifA is a free module overK. If it isn't, thenthe multiplication is still completely determined by its action on a set that spans A; however, the structure constantscan't be specied arbitrarily in this case, and knowing only the structure constants does not specify the algebra up toisomorphism.

1.6 Classication of low-dimensional algebrasTwo-dimensional, three-dimensional and four-dimensional unital associative algebras over the eld of complex num-bers were completely classied up to isomorphism by Eduard Study.[4]

There exist two two-dimensional algebras. Each algebra consists of linear combinations (with complex coecients)of two basis elements, 1 (the identity element) and a. According to the denition of an identity element,

1 1 = 1 ; 1 a = a ; a 1 = a :It remains to specify

aa = 1 for the rst algebra,aa = 0 for the second algebra.

There exist ve three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, 1(the identity element), a and b. Taking into account the denition of an identity element, it is sucient to specify

aa = a ; bb = b ; ab = ba = 0 for the rst algebra,aa = a ; bb = 0 ; ab = ba = 0 for the second algebra,aa = b ; bb = 0 ; ab = ba = 0 for the third algebra,aa = 1 ; bb = 0 ; ab = ba = b for the fourth algebra,aa = 0 ; bb = 0 ; ab = ba = 0 for the fth algebra.

The fourth algebra is non-commutative, others are commutative.

1.7 See also Cliord algebra Dierential algebra Geometric algebra Max-plus algebra Zariskis lemma Mutation (algebra)

1.8. NOTES 7

1.8 Notes[1] See also Hazewinkel et al. (2004). Algebras, rings and modules 1. p. 3.

[2] Joo B. Prolla, Approximation of vector valued functions, Elsevier, 1977, p. 65

[3] Richard D. Schafer, An Introduction to Nonassociative Algebras (1996) ISBN 0-486-68813-5 Gutenberg eText

[4] Study, E. (1890), "ber Systeme complexer Zahlen und ihre Anwendungen in der Theorie der Transformationsgruppen,Monatshefte fr Mathematik 1 (1): 283354, doi:10.1007/BF01692479

1.9 References Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhalovna Gubareni, Vladimir V. Kirichenko. Algebras,rings and modules. Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0

Chapter 2

Alternative algebra

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, onlyalternative. That is, one must have

x(xy) = (xx)y (yx)x = y(xx)

for all x and y in the algebra.Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as theoctonions. The sedenions, on the other hand, are not alternative.

2.1 The associatorAlternative algebras are so named because they are precisely the algebras for which the associator is alternating. Theassociator is a trilinear map given by

[x; y; z] = (xy)z x(yz)By denition a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left andright alternative identities for an algebra are equivalent to[1]

[x; x; y] = 0

[y; x; x] = 0:

Both of these identities together imply that the associator is totally skew-symmetric. That is,

[x(1); x(2); x(3)] = sgn()[x1; x2; x3]for any permutation . It follows that

[x; y; x] = 0

for all x and y. This is equivalent to the exible identity[2]

(xy)x = x(yx):

The associator of an alternative algebra is therefore alternating. Conversely, any algebra whose associator is alternatingis clearly alternative. By symmetry, any algebra which satises any two of:

8

2.2. EXAMPLES 9

left alternative identity: x(xy) = (xx)y right alternative identity: (yx)x = y(xx) exible identity: (xy)x = x(yx):

is alternative and therefore satises all three identities.An alternating associator is always totally skew-symmetric. The converse holds so long as the characteristic of thebase eld is not 2.

2.2 Examples Every associative algebra is alternative. The octonions form a non-associative alternative algebra, a normed division algebra of dimension 8 over thereal numbers.[3]

More generally, any octonion algebra is alternative.

2.3 PropertiesArtins theorem states that in an alternative algebra the subalgebra generated by any two elements is associative.[4]Conversely, any algebra for which this is true is clearly alternative. It follows that expressions involving only twovariables can be written without parenthesis unambiguously in an alternative algebra. A generalization of Artins the-orem states that whenever three elements x; y; z in an alternative algebra associate (i.e. [x; y; z] = 0 ) the subalgebragenerated by those elements is associative.A corollary of Artins theorem is that alternative algebras are power-associative, that is, the subalgebra generated bya single element is associative.[5] The converse need not hold: the sedenions are power-associative but not alternative.The Moufang identities

a(x(ay)) = (axa)y ((xa)y)a = x(aya) (ax)(ya) = a(xy)a

hold in any alternative algebra.[2]

In a unital alternative algebra, multiplicative inverses are unique whenever they exist. Moreover, for any invertibleelement x and all y one has

y = x1(xy):

This is equivalent to saying the associator [x1; x; y] vanishes for all such x and y . If x and y are invertible thenxy is also invertible with inverse (xy)1 = y1x1 . The set of all invertible elements is therefore closed undermultiplication and forms a Moufang loop. This loop of units in an alternative ring or algebra is analogous to the groupof units in an associative ring or algebra.Zorns theorem states that any nite-dimensional non-associative alternative algebra is a generalised octonion algebra.[6]

2.4 ApplicationsThe projective plane over any alternative division ring is a Moufang plane.The close relationship of alternative algebras and composition algebras was given by Guy Roos in 2008: He shows(page 162) the relation for an algebra A with unit element e and an involutive anti-automorphism a 7! a such thata + a* and aa* are on the line spanned by e for all a in A. Use the notation n(a) = aa*. Then if n is a non-singularmapping into the eld of A, and A is alternative, then (A,n) is a composition algebra.

10 CHAPTER 2. ALTERNATIVE ALGEBRA

2.5 See also Zorn ring Maltsev algebra

2.6 References[1] Schafer (1995) p.27

[2] Schafer (1995) p.28

[3] Conway, JohnHorton; Smith, DerekA. (2003). OnQuaternions and Octonions: Their Geometry, Arithmetic, and Symmetry.A. K. Peters. ISBN 1-56881-134-9. Zbl 1098.17001.

[4] Schafer (1995) p.29

[5] Schafer (1995) p.30

[6] Schafer (1995) p.56

Guy Roos (2008) Exceptional symmetric domains, 1: Cayley algebras, in Symmetries in Complex Analysisby Bruce Gilligan & Guy Roos, volume 468 of Contemporary Mathematics, American Mathematical Society.

Schafer, Richard D. (1995). An Introduction to Nonassociative Algebras. NewYork: Dover Publications. ISBN0-486-68813-5.

2.7 External links Zhevlakov, K.A. (2001), Alternative rings and algebras, in Hazewinkel, Michiel, Encyclopedia of Mathemat-ics, Springer, ISBN 978-1-55608-010-4

Chapter 3

Associative property

This article is about associativity in mathematics. For associativity in the central processing unit memory cache, seeCPU cache. For associativity in programming languages, see operator associativity.Associative and non-associative redirect here. For associative and non-associative learning, see Learning#Types.

In mathematics, the associative property[1] is a property of some binary operations. In propositional logic, associa-tivity is a valid rule of replacement for expressions in logical proofs.Within an expression containing two or more occurrences in a row of the same associative operator, the order inwhich the operations are performed does not matter as long as the sequence of the operands is not changed. That is,rearranging the parentheses in such an expression will not change its value. Consider the following equations:

(2 + 3) + 4 = 2 + (3 + 4) = 9

2 (3 4) = (2 3) 4 = 24:Even though the parentheses were rearranged, the values of the expressions were not altered. Since this holds truewhen performing addition and multiplication on any real numbers, it can be said that addition and multiplication ofreal numbers are associative operations.Associativity is not to be confused with commutativity, which addresses whether a b = b a.Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups andcategories) explicitly require their binary operations to be associative.However, many important and interesting operations are non-associative; some examples include subtraction, exponentiationand the vector cross product. In contrast to the theoretical counterpart, the addition of oating point numbers in com-puter science is not associative, and is an important source of rounding error.

3.1 DenitionFormally, a binary operation on a set S is called associative if it satises the associative law:

(x y) z = x (y z) for all x, y, z in S.

Here, is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbollike for the multiplication.

(xy)z = x(yz) = xyz for all x, y, z in S.

The associative law can also be expressed in functional notation thus: f(f(x, y), z) = f(x, f(y, z)).

11

12 CHAPTER 3. ASSOCIATIVE PROPERTY

A binary operation on the set S is associative when this diagram commutes. That is, when the two paths from SSS to S composeto the same function from SSS to S.

3.2 Generalized associative lawIf a binary operation is associative, repeated application of the operation produces the same result regardless how validpairs of parenthesis are inserted in the expression.[2] This is called the generalized associative law. For instance, aproduct of four elements may be written in ve possible ways:

1. ((ab)c)d

2. (ab)(cd)

3. (a(bc))d

4. a((bc)d)

5. a(b(cd))

If the product operation is associative, the generalized associative law says that all these formulas will yield the sameresult, making the parenthesis unnecessary. Thus the product can be written unambiguously as

abcd.

As the number of elements increases, the number of possible ways to insert parentheses grows quickly, but theyremain unnecessary for disambiguation.

3.3 ExamplesSome examples of associative operations include the following.

The concatenation of the three strings hello, " ", world can be computed by concatenating the rst twostrings (giving hello ") and appending the third string (world), or by joining the second and third string(giving " world) and concatenating the rst string (hello) with the result. The two methods produce thesame result; string concatenation is associative (but not commutative).

In arithmetic, addition and multiplication of real numbers are associative; i.e.,

3.3. EXAMPLES 13

(((ab)c)d)e

((ab)c)(de)

((ab)(cd))e

((a(bc))d)e

(ab)(c(de))

(a(bc))(de)

(ab)((cd)e)

(a(b(cd)))e

a(b(c(de)))

a((bc)(de))

a(b((cd)e))

a(((bc)d)e)

a((b(cd))e)

(a((bc)d))e

In the absence of the associative property, ve factors a, b, c, d, e result in a Tamari lattice of order four, possibly dierent products.

(x+ y) + z = x+ (y + z) = x+ y + z(x y)z = x(y z) = x y z

for all x; y; z 2 R:

Because of associativity, the grouping parentheses can be omitted without ambiguity.

14 CHAPTER 3. ASSOCIATIVE PROPERTY

(x + z+ y)

x + z)+ (y=

The addition of real numbers is associative.

Addition and multiplication of complex numbers and quaternions are associative. Addition of octonions is alsoassociative, but multiplication of octonions is non-associative.

The greatest common divisor and least common multiple functions act associatively.

gcd(gcd(x; y); z) = gcd(x; gcd(y; z)) = gcd(x; y; z)lcm(lcm(x; y); z) = lcm(x; lcm(y; z)) = lcm(x; y; z)

for all x; y; z 2 Z:

Taking the intersection or the union of sets:

(A \B) \ C = A \ (B \ C) = A \B \ C(A [B) [ C = A [ (B [ C) = A [B [ C

for all sets A;B;C:

IfM is some set and S denotes the set of all functions fromM toM, then the operation of functional compositionon S is associative:

(f g) h = f (g h) = f g h for all f; g; h 2 S:

Slightly more generally, given four sets M, N, P and Q, with h: M to N, g: N to P, and f: P to Q, then

(f g) h = f (g h) = f g h

as before. In short, composition of maps is always associative.

Consider a set with three elements, A, B, and C. The following operation:

is associative. Thus, for example, A(BC)=(AB)C = A. This operation is not commutative.

Because matrices represent linear transformation functions, with matrix multiplication representing functionalcomposition, one can immediately conclude that matrix multiplication is associative.

3.4. PROPOSITIONAL LOGIC 15

3.4 Propositional logic

3.4.1 Rule of replacementIn standard truth-functional propositional logic, association,[3][4] or associativity[5] are two valid rules of replacement.The rules allow one to move parentheses in logical expressions in logical proofs. The rules are:

(P _ (Q _R)), ((P _Q) _R)

and

(P ^ (Q ^R)), ((P ^Q) ^R);

where ", " is a metalogical symbol representing can be replaced in a proof with.

3.4.2 Truth functional connectivesAssociativity is a property of some logical connectives of truth-functional propositional logic. The following logicalequivalences demonstrate that associativity is a property of particular connectives. The following are truth-functionaltautologies.Associativity of disjunction:

(P _ (Q _R))$ ((P _Q) _R)

((P _Q) _R)$ (P _ (Q _R))Associativity of conjunction:

((P ^Q) ^R)$ (P ^ (Q ^R))

(P ^ (Q ^R))$ ((P ^Q) ^R)Associativity of equivalence:

((P $ Q)$ R)$ (P $ (Q$ R))

(P $ (Q$ R))$ ((P $ Q)$ R)

3.5 Non-associativityA binary operation on a set S that does not satisfy the associative law is called non-associative. Symbolically,

(x y) z 6= x (y z) for some x; y; z 2 S:

For such an operation the order of evaluation does matter. For example:

Subtraction

(5 3) 2 6= 5 (3 2)

16 CHAPTER 3. ASSOCIATIVE PROPERTY

Division

(4/2)/2 6= 4/(2/2)

Exponentiation

2(12) 6= (21)2

Also note that innite sums are not generally associative, for example:

(1 1) + (1 1) + (1 1) + (1 1) + (1 1) + (1 1) + : : : = 0

whereas

1 + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + (1 + : : : = 1

The study of non-associative structures arises from reasons somewhat dierent from the mainstream of classicalalgebra. One area within non-associative algebra that has grown very large is that of Lie algebras. There the associativelaw is replaced by the Jacobi identity. Lie algebras abstract the essential nature of innitesimal transformations, andhave become ubiquitous in mathematics.There are other specic types of non-associative structures that have been studied in depth; these tend to come fromsome specic applications or areas such as combinatorial mathematics. Other examples are Quasigroup, Quasield,Non-associative ring, Non-associative algebra and Commutative non-associative magmas.

3.5.1 Nonassociativity of oating point calculation

In mathematics, addition and multiplication of real numbers is associative. By contrast, in computer science, theaddition and multiplication of oating point numbers is not associative, as rounding errors are introduced whendissimilar-sized values are joined together.[6]

To illustrate this, consider a oating point representation with a 4-bit mantissa:(1.000220 + 1.000220) + 1.000224 = 1.000221 + 1.000224 = 1.0012241.000220 + (1.000220 + 1.000224) = 1.000220 + 1.000224 = 1.000224

Even though most computers compute with a 24 or 53 bits of mantissa,[7] this is an important source of roundingerror, and approaches such as the Kahan Summation Algorithm are ways to minimise the errors. It can be especiallyproblematic in parallel computing.[8] [9]

3.5.2 Notation for non-associative operations

Main article: Operator associativity

In general, parentheses must be used to indicate the order of evaluation if a non-associative operation appears morethan once in an expression. However, mathematicians agree on a particular order of evaluation for several commonnon-associative operations. This is simply a notational convention to avoid parentheses.A left-associative operation is a non-associative operation that is conventionally evaluated from left to right, i.e.,

x y z = (x y) zw x y z = ((w x) y) zetc.

9=; for all w; x; y; z 2 Swhile a right-associative operation is conventionally evaluated from right to left:

3.5. NON-ASSOCIATIVITY 17

x y z = x (y z)w x y z = w (x (y z))etc.

9=; for all w; x; y; z 2 SBoth left-associative and right-associative operations occur. Left-associative operations include the following:

Subtraction and division of real numbers:

x y z = (x y) z for all x; y; z 2 R;x/y/z = (x/y)/z for all x; y; z 2 R with y 6= 0; z 6= 0:

Function application:

(f x y) = ((f x) y)

This notation can be motivated by the currying isomorphism.

Right-associative operations include the following:

Exponentiation of real numbers:

xyz

= x(yz):

The reason exponentiation is right-associative is that a repeated left-associative exponentiation operationwould be less useful. Multiple appearances could (and would) be rewritten with multiplication:

(xy)z = x(yz):

Function denition

Z! Z! Z = Z! (Z! Z)x 7! y 7! x y = x 7! (y 7! x y)

Using right-associative notation for these operations can be motivated by the Curry-Howard correspon-dence and by the currying isomorphism.

Non-associative operations for which no conventional evaluation order is dened include the following.

Taking the Cross product of three vectors:

~a (~b ~c) 6= (~a~b) ~c for some ~a;~b;~c 2 R3

Taking the pairwise average of real numbers:

(x+ y)/2 + z

26= x+ (y + z)/2

2for all x; y; z 2 R with x 6= z:

Taking the relative complement of sets (AnB)nC is not the same as An(BnC) . (Compare material nonim-plication in logic.)

18 CHAPTER 3. ASSOCIATIVE PROPERTY

3.6 See also Lights associativity test A semigroup is a set with a closed associative binary operation. Commutativity and distributivity are two other frequently discussed properties of binary operations. Power associativity, alternativity and N-ary associativity are weak forms of associativity.

3.7 References[1] Thomas W. Hungerford (1974). Algebra (1st ed.). Springer. p. 24. ISBN 0387905189. Denition 1.1 (i) a(bc) = (ab)c

for all a, b, c in G.

[2] Durbin, John R. (1992). Modern Algebra: an Introduction (3rd ed.). New York: Wiley. p. 78. ISBN 0-471-51001-7. Ifa1; a2; : : : ; an (n 2) are elements of a set with an associative operation, then the product a1a2 : : : an is unambiguous;this is, the same element will be obtained regardless of how parentheses are inserted in the product

[3] Moore and Parker

[4] Copi and Cohen

[5] Hurley

[6] Knuth, Donald, The Art of Computer Programming, Volume 3, section 4.2.2

[7] IEEEComputer Society (August 29, 2008). IEEE Standard for Floating-Point Arithmetic. IEEE. doi:10.1109/IEEESTD.2008.4610935.ISBN 978-0-7381-5753-5. IEEE Std 754-2008.

[8] Villa, Oreste; Chavarra-mir, Daniel; Gurumoorthi, Vidhya; Mrquez, Andrs; Krishnamoorthy, Sriram, Eects of Floating-Point non-Associativity on Numerical Computations on Massively Multithreaded Systems (PDF), retrieved 2014-04-08

[9] Goldberg, David, What Every Computer Scientist ShouldKnowAbout Floating Point Arithmetic (PDF),ACMComputingSurveys 23 (1): 548, doi:10.1145/103162.103163, retrieved 2014-04-08

Chapter 4

Flexible algebra

In mathematics, a exible binary operation is a binary operation that satises the equation

a (b a) = (a b) a:

for any two elements a and b of an algebraic structure.Every commutative or associative operation is exible, so the exible identity becomes important for binary op-erations that are neither commutative nor associative, e.g. for the multiplication of sedenions, which are not evenalternative.

4.1 ExamplesThe following classes of algebra are exible:

Alternative algebra Lie algebras Jordan algebras Okubo algebras

4.2 See also Zorn ring Maltsev algebra

4.3 References Schafer, Richard D. (1995) [1966]. An introduction to non-associative algebras. Dover Publications. ISBN0-486-68813-5. Zbl 0145.25601.

19

Chapter 5

Non-associative algebra

This article is about a particular structure known as a non-associative algebra. For non-associativity in general, seeNon-associativity.

A non-associative algebra[1] (or distributive algebra) over a eld (or a commutative ring) K is a K-vector space (ormore generally a module[2]) A equipped with a K-bilinear map A A A which establishes a binary multiplicationoperation on A. Since it is not assumed that the multiplication is associative, using parentheses to indicate the orderof multiplications is necessary. For example, the expressions (ab)(cd), (a(bc))d and a(b(cd)) may all yield dierentanswers.While this use of non-associative means that associativity is not assumed, it does not mean that associativity is disal-lowed. In other words, non-associative means not necessarily associative, just as noncommutative means notnecessarily commutative for noncommutative rings.An algebra is unital or unitary if it has an identity element I with Ix = x = xI for all x in the algebra.The nonassociative algebra structure of A may be studied by associating it with other associative algebras which aresubalgebra of the full algebra of K-endomorphisms of A as a K-vector space. Two such are the derivation algebraand the (associative) enveloping algebra, the latter being in a sense the smallest associative algebra containing A".

5.1 Algebras satisfying identitiesRing-like structures with two binary operations and no other restrictions are a broad class, one which is too generalto study. For this reason, the best-known kinds of non-associative algebras satisfy identities which simplify multipli-cation somewhat. These include the following identities.In the list, x, y and z denote arbitrary elements of an algebra.

Associative: (xy)z = x(yz). Commutative: xy = yx. Anticommutative:[3] xy = yx.[4]

Jacobi identity:[3][5] (xy)z + (yz)x + (zx)y = 0. Jordan identity:[6][7] (xy)x2 = x(yx2). Power associative:[8][9][10] For all x, any three nonnegative powers of x associate. That is if a, b and c arenonnegative powers of x, then a(bc) = (ab)c. This is equivalent to saying that xm xn = xn+m for all non-negativeintegers m and n.

Alternative:[11][12][13] (xx)y = x(xy) and (yx)x = y(xx). Flexible:[14][15] x(yx) = (xy)x. Elastic:[16] Flexible and (xy)(xx) = x(y(xx)), x(xx)y = (xx)(xy).

20

5.2. EXAMPLES 21

These properties are related by

1. associative implies alternative implies power associative;2. associative implies Jordan identity implies power associative;3. Each of the properties associative, commutative, anticommutative, Jordan identity, and Jacobi identity individ-

ually imply exible.[14][15]

4. For a eld with characteristic not two, being both commutative and anticommutative implies the algebra is just{0}.

5.1.1 AssociatorMain article: Associator

The associator on A is the K-multilinear map [; ; ] : AAA! A given by

[x; y; z] = (xy)z x(yz):It measures the degree of nonassociativity of A , and can be used to conveniently express some possible identitiessatised by A.

Associative: the associator is identically zero; Alternative: the associator is alternating, interchange of any two terms changes the sign; Flexible: [x; y; x] = 0 ; Jordan: [x; y; x2] = 0 .[17]

The nucleus is the set of elements that associate with all others:[18] that is, the n in A such that

[n;A;A] = [A;n;A] = [A;A; n] = f0g :

5.2 Examples Euclidean space R3 with multiplication given by the vector cross product is an example of an algebra which isanticommutative and not associative. The cross product also satises the Jacobi identity.

Lie algebras are algebras satisfying anticommutativity and the Jacobi identity. Algebras of vector elds on a dierentiable manifold (if K is R or the complex numbers C) or an algebraicvariety (for general K);

Jordan algebras are algebras which satisfy the commutative law and the Jordan identity.[7]

Every associative algebra gives rise to a Lie algebra by using the commutator as Lie bracket. In fact every Liealgebra can either be constructed this way, or is a subalgebra of a Lie algebra so constructed.

Every associative algebra over a eld of characteristic other than 2 gives rise to a Jordan algebra by dening anew multiplication x*y = (1/2)(xy + yx). In contrast to the Lie algebra case, not every Jordan algebra can beconstructed this way. Those that can are called special.

Alternative algebras are algebras satisfying the alternative property. Themost important examples of alternativealgebras are the octonions (an algebra over the reals), and generalizations of the octonions over other elds. Allassociative algebras are alternative. Up to isomorphism, the only nite-dimensional real alternative, divisionalgebras (see below) are the reals, complexes, quaternions and octonions.

22 CHAPTER 5. NON-ASSOCIATIVE ALGEBRA

Power-associative algebras, are those algebras satisfying the power-associative identity. Examples include allassociative algebras, all alternative algebras, Jordan algebras, and the sedenions.

The hyperbolic quaternion algebra overR, whichwas an experimental algebra before the adoption ofMinkowskispace for special relativity.

More classes of algebras:

Graded algebras. These include most of the algebras of interest to multilinear algebra, such as the tensoralgebra, symmetric algebra, and exterior algebra over a given vector space. Graded algebras can be generalizedto ltered algebras.

Division algebras, in which multiplicative inverses exist. The nite-dimensional alternative division algebrasover the eld of real numbers have been classied. They are the real numbers (dimension 1), the complexnumbers (dimension 2), the quaternions (dimension 4), and the octonions (dimension 8). The quaternions andoctonions are not commutative. Of these algebras, all are associative except for the octonions.

Quadratic algebras, which require that xx = re + sx, for some elements r and s in the ground eld, and e a unitfor the algebra. Examples include all nite-dimensional alternative algebras, and the algebra of real 2-by-2matrices. Up to isomorphism the only alternative, quadratic real algebras without divisors of zero are the reals,complexes, quaternions, and octonions.

The CayleyDickson algebras (where K is R), which begin with: C (a commutative and associative algebra); the quaternions H (an associative algebra); the octonions (an alternative algebra); the sedenions (a power-associative algebra, like all of the Cayley-Dickson algebras).

The Poisson algebras are considered in geometric quantization. They carry two multiplications, turning theminto commutative algebras and Lie algebras in dierent ways.

Genetic algebras are non-associative algebras used in mathematical genetics.

5.3 Free non-associative algebraThe free non-associative algebra on a set X over a eld K is dened as the algebra with basis consisting of all non-associative monomials, nite formal products of elements of X retaining parentheses. The product of monomials u,v is just (u)(v). The algebra is unital if one takes the empty product as a monomial.[19]

Kurosh proved that every subalgebra of a free non-associative algebra is free.[20]

5.4 Associated algebrasAn algebra A over a eld K is in particular a K-vector space and so one can consider the associative algebra EndK(A)of K-linear vector space endomorphism of A. We can associate to the algebra structure on A two subalgebras ofEndK(A), the derivation algebra and the (associative) enveloping algebra.

5.4.1 Derivation algebraA derivation on A is a map D with the property

D(x y) = D(x) y + x D(y) :The derivations on A form a subspace DerK(A) in EndK(A). The commutator of two derivations is again a derivation,so that the Lie bracket gives DerK(A) a structure of Lie algebra.[21]

5.5. SEE ALSO 23

5.4.2 Enveloping algebraThere are linear maps L and R attached to each element a of an algebra A:[22]

L(a) : x 7! ax; R(a) : x 7! xa :

The associative enveloping algebra or multiplication algebra of A is the associative algebra generated by the left andright linear maps.[17][23] The centroid of A is the centraliser of the enveloping algebra in the endomorphism algebraEndK(A). An algebra is central if its centroid consists of the K-scalar multiples of the identity.[10]

Some of the possible identities satised by non-associative algebras may be conveniently expressed in terms of thelinear maps:[24]

Commutative: each L(a) is equal to the corresponding R(a); Associative: any L commutes with any R; Flexible: every L(a) commutes with the corresponding R(a); Jordan: every L(a) commutes with R(a2); Alternative: every L(a)2 = L(a2) and similarly for the right.

The quadratic representation Q is dened by[25]

Q(a) : x 7! 2a (a x) (a a) x

or equivalently

Q(a) = 2L2(a) L(a2) :

5.5 See also List of algebras Commutative non-associative magmas, which give rise to non-associative algebras

5.6 Notes[1] Schafer 1966, Chapter 1.

[2] Schafer 1966, pp.1.

[3] Schafer (1995) p.3

[4] This is always implied by the identity xx = 0 for all x, and the converse holds for elds of characteristic other than two.

[5] Okubo (2005) p.12

[6] Schafer (1995) p.91

[7] Okubo (2005) p.13

[8] Schafer (1995) p.30

[9] Okubo (2005) p.17

[10] Knus et al (1998) p.451

24 CHAPTER 5. NON-ASSOCIATIVE ALGEBRA

[11] Schafer (1995) p.5

[12] Okubo (2005) p.18

[13] McCrimmon (2004) p.153

[14] Schafer (1995) p.28

[15] Okubo (2005) p.16

[16] Rosenfeld, Boris (1997). Geometry of Lie groups. Mathematics and its Applications 393. Dordrecht: Kluwer AcademicPublishers. p. 91. ISBN 0792343905. Zbl 0867.53002.

[17] Schafer (1995) p.14

[18] McCrimmon (2004) p.56

[19] Rowen, Louis Halle (2008). Graduate Algebra: Noncommutative View. Graduate studies in mathematics. AmericanMathematical Society. p. 321. ISBN 0-8218-8408-5.

[20] Kurosh, A.G. (1947). Non-associative algebras and free products of algebras. Mat. Sbornik 20 (62): 237262. MR20986. Zbl 0041.16803.

[21] Schafer (1995) p.4

[22] Okubo (2004) p.24

[23] Albert, A. Adrian (2003) [1939]. Structure of algebras. American Mathematical Society Colloquium Publ. 24 (Correctedreprint of the revised 1961 ed.). New York: American Mathematical Society. p. 113. ISBN 0-8218-1024-3. Zbl0023.19901.

[24] McCrimmon (2004) p.57

[25] Koecher (1999) p.57

5.7 References Herstein, I. N., ed. (2011) [1965], Some Aspects of Ring Theory: Lectures given at a Summer School of the Cen-tro Internazionale Matematico Estivo (C.I.M.E.) held in Varenna (Como), Italy, August 23-31, 1965, C.I.M.E.Summer Schools 37 (reprint ed.), Springer-Verlag, ISBN 3642110363

Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions,Colloquium Publications 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, ISBN0-8218-0904-0, Zbl 0955.16001

Koecher, Max (1999), Krieg, Aloys; Walcher, Sebastian, eds., The Minnesota notes on Jordan algebras andtheir applications, Lecture Notes in Mathematics 1710, Berlin: Springer-Verlag, ISBN 3-540-66360-6, Zbl1072.17513

McCrimmon, Kevin (2004), A taste of Jordan algebras, Universitext, Berlin, New York: Springer-Verlag,doi:10.1007/b97489, ISBN 978-0-387-95447-9, MR 2014924, Zbl 1044.17001, Errata

Okubo, Susumu (2005) [1995], Introduction to Octonion and Other Non-Associative Algebras in Physics, Mon-trollMemorial Lecture Series inMathematical Physics 2, CambridgeUniversity Press, doi:10.1017/CBO9780511524479,ISBN 0-521-01792-0, Zbl 0841.17001

Schafer, Richard D. (1995) [1966], An Introduction to Nonassociative Algebras, Dover, ISBN 0-486-68813-5,Zbl 0145.25601

Chapter 6

Octonion

In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capitalletter O, using boldface O or blackboard bold O . There are only four such algebras, the other three being thereal numbers R, the complex numbers C, and the quaternions H. The octonions are the largest such algebra, witheight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They arenoncommutative and nonassociative, but satisfy a weaker form of associativity, namely they are alternative.Octonions are not as well known as the quaternions and complex numbers, which are much more widely studiedand used. Despite this, they have some interesting properties and are related to a number of exceptional structuresin mathematics, among them the exceptional Lie groups. Additionally, octonions have applications in elds such asstring theory, special relativity, and quantum logic.The octonions were discovered in 1843 by John T. Graves, inspired by his friend William Hamilton's discovery ofquaternions. Graves called his discovery octaves, and mentioned them in a letter to Hamilton dated 16 December1843, but his rst publication of his result in (Graves 1845) was slightly later than Cayleys article on them. Theoctonions were discovered independently by Arthur Cayley[1] and are sometimes referred to as Cayley numbers orthe Cayley algebra. Hamilton (1848) described the early history of Graves discovery.

6.1 DenitionThe octonions can be thought of as octets (or 8-tuples) of real numbers. Every octonion is a real linear combinationof the unit octonions:

fe0; e1; e2; e3; e4; e5; e6; e7g;

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

x = x0e0 + x1e1 + x2e2 + x3e3 + x4e4 + x5e5 + x6e6 + x7e7;

with real coecients {xi}.Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coe-cients, like quaternions. Multiplication is more complex. Multiplication is distributive over addition, so the productof two octonions can be calculated by summing the product of all the terms, again like quaternions. The product ofeach term can be given by multiplication of the coecients and a multiplication table of the unit octonions, like thisone:[2]

Most o-diagonal elements of the table are antisymmetric, making it almost a skew-symmetric matrix except for theelements on the main diagonal, as well as the row and column for which e0 is an operand.The table can be summarized by the relations:[3]

25

26 CHAPTER 6. OCTONION

eiej = ije0 + "ijkek;

where "ijk is a completely antisymmetric tensor with value +1 when ijk = 123, 145, 176, 246, 257, 347, 365, and:

eie0 = e0ei = ei; e0e0 = e0;

with e0 the scalar element, and i, j, k = 1 ... 7.The above denition though is not unique, but is only one of 480 possible denitions for octonion multiplication withe0 = 1. The others can be obtained by permuting and changing the signs of the non-scalar basis elements. The 480dierent algebras are isomorphic, and there is rarely a need to consider which particular multiplication rule is used.Each of these 480 denitions is invariant up to signs under some 7-cycle of the points (1234567), and for each 7-cyclethere are four denitions, diering by signs and reversal of order. A common choice is to use the denition invariantunder the 7-cycle (1234567) with e1e2 = e4 as it is particularly easy to remember the multiplication. A variation ofthis sometimes used is to label the elements of the basis by the elements , 0, 1, 2, ..., 6, of the projective line overthe nite eld of order 7. The multiplication is then given by e = 1 and e1e2 = e4, and all expressions obtained fromthis by adding a constant (mod 7) to all subscripts: in other words using the 7 triples (124) (235) (346) (450) (561)(602) (013). These are the nonzero codewords of the quadratic residue code of length 7 over the eld of 2 elements.There is a symmetry of order 7 given by adding a constant mod 7 to all subscripts, and also a symmetry of order 3given by multiplying all subscripts by one of the quadratic residues 1, 2, 4 mod 7.[4][5]

The multiplication table can be given in terms of the following 7 quaternionic triples (omitting the identity element):(Ijk), (iJk), (ijK), (IJK), (Iim), (Jjm), (Kkm) in which the lowercase items are vectors (mathematics and physics) andthe uppercase ones are bivectors.

6.1.1 CayleyDickson constructionA more systematic way of dening the octonions is via the CayleyDickson construction. Just as quaternions canbe dened as pairs of complex numbers, the octonions can be dened as pairs of quaternions. Addition is denedpairwise. The product of two pairs of quaternions (a, b) and (c, d) is dened by

(a; b)(c; d) = (ac db; da+ bc)

where z denotes the conjugate of the quaternion z. This denition is equivalent to the one given above when theeight unit octonions are identied with the pairs

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

6.1.2 Fano plane mnemonicA convenient mnemonic for remembering the products of unit octonions is given by the diagram at the right, whichrepresents the multiplication table of Cayley and Graves.[2][7] This diagram with seven points and seven lines (thecircle through 1, 2, and 3 is considered a line) is called the Fano plane. The lines are oriented. The seven pointscorrespond to the seven standard basis elements of Im(O) (see denition below). Each pair of distinct points lies ona 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 specied 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,

6.1. DEFINITION 27

e1=I

e2=J

e3=IJ

e4=L

e5=IL

e7=IJL

e6=JL

A mnemonic for the products of the unit octonions.[6]

ei2 = 1 for each point in the diagram

completely denes the multiplicative structure of the octonions. Each of the seven lines generates a subalgebra of Oisomorphic to the quaternions H.

6.1.3 Conjugate, norm, and inverseThe conjugate of an octonion

x = x0 e0 + x1 e1 + x2 e2 + x3 e3 + x4 e4 + x5 e5 + x6 e6 + x7 e7

is given by

x = x0 e0 x1 e1 x2 e2 x3 e3 x4 e4 x5 e5 x6 e6 x7 e7:Conjugation is an involution of O and satises (xy)* = y* x* (note the change in order).The real part of x is given by

x+ x

2= x0 e0

28 CHAPTER 6. OCTONION

and the imaginary part by

x x2

= x1 e1 + x2 e2 + x3 e3 + x4 e4 + x5 e5 + x6 e6 + x7 e7:

The set of all purely imaginary octonions span a 7 dimension subspace of O, denoted Im(O).Conjugation of octonions satises the equation

x = 16(x+ (e1x)e1 + (e2x)e2 + (e3x)e3 + (e4x)e4 + (e5x)e5 + (e6x)e6 + (e7x)e7):

The product of an octonion with its conjugate, x* x = x x*, is always a nonnegative real number:

xx = x20 + x21 + x

22 + x

23 + x

24 + x

25 + x

26 + x

27:

Using this the norm of an octonion can be dened, as

kxk = pxx:This norm agrees with the standard Euclidean norm on R8.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

x1 =x

kxk2 :

It satises x x1 = x1 x = 1.

6.2 PropertiesOctonionic multiplication is neither commutative:

eiej = ejei 6= ejei if i; j are distinct and non-zero,

nor associative:

(eiej)ek = ei(ejek) 6= ei(ejek) if i; j; k are distinct, non-zero or if eiej 6= ek .

The octonions do satisfy a weaker form of associativity: they are alternative. This means that the subalgebra generatedby any two elements is associative. Actually, one can show that the subalgebra generated by any two elements of Ois isomorphic to R, C, or H, all of which are associative. Because of their non-associativity, octonions don't havematrix representations, unlike quaternions.The octonions do retain one important property shared by R, C, and H: the norm on O satises

kxyk = kxkkykThis implies that the octonions form a nonassociative normed division algebra. The higher-dimensional algebrasdened by the CayleyDickson construction (e.g. the sedenions) all fail to satisfy this property. They all have zerodivisors.Wider number systems exist which have a multiplicative modulus (e.g. 16 dimensional conic sedenions). Theirmodulus is dened dierently from their norm, and they also contain zero divisors.It turns out that the only normed division algebras over the reals are R, C, H, and O. These four algebras also formthe only alternative, nite-dimensional division algebras over the reals (up to isomorphism).Not being associative, the nonzero elements of O do not form a group. They do, however, form a loop, indeed aMoufang loop.

6.3. INTEGRAL OCTONIONS 29

6.2.1 Commutator and cross productThe commutator of two octonions x and y is given by

[x; y] = xy yx:

This is antisymmetric and imaginary. If it is considered only as a product on the imaginary subspace Im(O) it denesa product on that space, the seven-dimensional cross product, given by

x y = 12(xy yx):

Like the cross product in three dimensions this is a vector orthogonal to x and y with magnitude

kx yk = kxkkyk sin :

But like the octonion product it is not uniquely dened. Instead there are many dierent cross products, each onedependent on the choice of octonion product.[8]

6.2.2 AutomorphismsAn automorphism, A, of the octonions is an invertible linear transformation of O which satises

A(xy) = A(x)A(y):

The set of all automorphisms ofO forms a group called G2.[9] The group G2 is a simply connected, compact, real Liegroup of dimension 14. This group is the smallest of the exceptional Lie groups and is isomorphic to the subgroupof 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.

6.2.3 IsotopiesAn 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=cthis is the same as an automorphism. The isotopy group of an algebra is the group of all isotopies, which containsthe group of automorphisms as a subgroup.The isotopy group of the octonions is the group Spin8(R), with a, b, and c acting as the three 8-dimensional representations.[10]The subgroup of elements where c xes the identity is the subgroup Spin7(R), and the subgroup where a, b, and c allx the identity is the automorphism group G2.

6.3 Integral octonionsThere are several natural ways to choose an integral form of the octonions. The simplest is just to take the octonionswhose coordinates are integers. This gives a nonassociative algebra over the integers called the Gravesian octonions.However it is not a maximal order, and there are exactly 7 maximal orders containing it. These 7 maximal orders areall equivalent under automorphisms. The phrase integral octonions usually refers to a xed choice of one of theseseven orders.These maximal orders were constructed by Kirmse (1925), Dickson and Bruck as follows. Label the 8 basis vectorsby the points of the projective plane over the eld with 7 elements. First form the Kirmse integers : these consistof octonions whose coordinates are integers or half integers, and that are half odd integers on one of the 16 sets

30 CHAPTER 6. OCTONION

(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 eld of 2 elements, given by , (124) and its imagesunder adding a constant mod 7, and the complements of these 8 sets. (Kirmse incorrectly claimed that these forma maximal order, so thought there were 8 maximal orders rather than 7, but as Coxeter (1946) pointed out they arenot closed under multiplication; this mistake occurs in several published papers.) Then switch innity and any othercoordinate; this gives a maximal order. There are 7 ways to do this, giving 7 maximal orders, which are all equivalentunder cyclic permutations of the 7 coordinates 0123456.The Kirmse integers and the 7 maximal orders are all isometric to the E8 lattice rescaled by a factor of 1/2. Inparticular there are 240 elements of minimum nonzero norm 1 in each of these orders, forming a Moufang loop oforder 240.The integral octonions have a division with remainder property: given integral octonions a and b0, we can nd qand 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 principalideals 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 becausethe 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 aproduct of irreducible octonions. More precisely an integral octonion of norm mn can be written as a product ofintegral octonions of norms m and n.The automorphism group of the integral octonions is the group G2(F2) of order 12096, which has a simple subgroupof index 2 isomorphic to the unitary group 2A2(32). The isotopy group of the integral octonions is the perfect doublecover of the group of rotations of the E8 lattice.

6.4 See also Composition algebra Octonion algebra Okubo algebra Spin(8) Split-octonions Triality

6.5 Notes[1] Arthur Cayley (1845)

[2] This table is due to Arthur Cayley (1845) and John T. Graves (1843). See G Gentili, C Stoppato, DC Struppa and FVlacci (2009), Recent developments for regular functions of a hypercomplex variable, in Irene Sabadini, M Shapiro, FSommen, Hypercomplex analysis (Conference on quaternionic and Cliord analysis; proceedings ed.), Birkhuser, p. 168,ISBN 978-3-7643-9892-7

[3] Lev Vasilevitch Sabinin, Larissa Sbitneva, I. P. Shestakov (2006), "17.2 Octonion algebra and its regular bimodule rep-resentation, Non-associative algebra and its applications, CRC Press, p. 235, ISBN 0-8247-2669-3

[4] Rafa Abamowicz, Pertti Lounesto, Josep M. Parra (1996), " Four ocotonionic basis numberings, Cliord algebras withnumeric and symbolic computations, Birkhuser, p. 202, ISBN 0-8176-3907-1

[5] Jrg Schray, Corinne A. Manogue (1996), Octonionic representations of Cliord algebras and triality, Foundations ofphysics (Springer) 26 (Number 1/January): 1770, doi:10.1007/BF02058887. Available as ArXive preprint Figure 1 islocated here.

6.6. REFERENCES 31

[6] (Baez 2002, p. 6)

[7] Tevian Dray & Corinne A Manogue (2004), Chapter 29: Using octonions to describe fundamental particles, in PerttiLounesto, Rafa Abamowicz, Cliord algebras: applications to mathematics, physics, and engineering, Birkhuser, p. 452,ISBN 0-8176-3525-4 Figure 29.1: Representation of multiplication table on projective plane.

[8] (Baez 2002, pp. 3738)

[9] (Conway & Smith 2003, Chapter 8.6)

[10] (Conway & Smith 2003, Chapter 8)

6.6 References Baez, John C. (2002). The Octonions. Bulletin of the American Mathematical Society 39 (2): 145205.arXiv:math/0105155v4. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979. MR 1886087.

Baez, John C. (2005). Errata for The Octonions" (PDF). Bulletin of the American Mathematical Society 42(2): 213213. doi:10.1090/S0273-0979-05-01052-9.

Cayley, Arthur (1845), On Jacobis elliptic functions, in reply to the Rev..; and on quaternions, Philos. Mag.26: 208211, doi:10.1080/14786444508645107. Appendix reprinted in The Collected Mathematical Papers,Johnson Reprint Co., New York, 1963, p. 127.

Conway, John Horton; Smith, Derek A. (2003), On Quaternions and Octonions: Their Geometry, Arithmetic,and Symmetry, A. K. Peters, Ltd., ISBN 1-56881-134-9, Zbl 1098.17001. (Review).

Coxeter, H. S. M. (1946), Integral Cayley numbers., Duke Math. J. 13: 561578, doi:10.1215/s0012-7094-46-01347-6, MR 0019111

Freudenthal, Hans (1985) [1951], Oktaven, Ausnahmegruppen und Oktavengeometrie, Geom. Dedicata 19(1): 763, doi:10.1007/BF00233101, MR 0797151

Graves (1845), On a Connection between the General Theory of Normal Couples and the Theory of CompleteQuadratic Functions of Two Variables, Philos. Mag. 26: 315320, doi:10.1080/14786444508645136

Hamilton (1848), Note, by Sir W. R. Hamilton, respecting the researches of John T. Graves, Esq., Trans.Roy. Irish Acad. 21: 338341

Kirmse (1925), Ber. Verh. Schs. Akad. Wiss. Leipzig. Math. Phys. Kl. 76: 6382 Missing or empty |title=(help)

van der Blij, F. (1960/1961), History of the octaves., Simon Stevin 34: 106125, MR 0130283 Check datevalues in: |date= (help)

6.7 External links Hazewinkel, Michiel, ed. (2001), Cayley numbers, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Octonions and the Fano Plane Mnemonic (video demonstration) Wilson, R. A. (2008), Octonions (PDF), Pure Mathematics Seminar notes

32 CHAPTER 6. OCTONION

6.8 Text and image sources, contributors, and licenses6.8.1 Text

Algebra over a eld Source: https://en.wikipedia.org/wiki/Algebra_over_a_field?oldid=669778616 Contributors: AxelBoldt, Zundark,Toby Bartels, FvdP, Netesq, Michael Hardy, TakuyaMurata, GTBacchus, Jordi Burguet Castell, Loren Rosen, Charles Matthews, Dys-prosia, Jitse Niesen, VeryVerily, Phys, Robbot, Robinh, Ruakh, Tobias Bergemann, Marc Venot, Giftlite, Fropu, Dratman, Waltpohl,Wmahan, DefLog~enwiki, Alberto da Calvairate~enwiki, Gauss, TedPavlic, Guanabot, Paul August, Rgdboer, EmilJ, Mdd, Tsirel,HasharBot~enwiki, Keenan Pepper, Mlm42, Drbreznjev, Oleg Alexandrov, Woohookitty, Linas, Julien Tuerlinckx, MFH, Marudub-shinki, Magister Mathematicae, Jake Wartenberg, Staecker, FlaBot, Eubot, Mathbot, Joseluisap~enwiki, Mhking, YurikBot, RobotE,Ihope127, Welsh, Crasshopper, Dan131m, Blurble, SmackBot, RDBury, Amcbride, Reedy, Bluebot, Silly rabbit, Vanished User 0001,Ccero, Wiki me, Henning Makholm, Lambiam, Khazar, Michael Kinyon, WhiteHatLurker, Dicklyon, Rschwieb, CRGreathouse, Cm-drObot, Anupam, Rlupsa, Hammerhorn~enwiki, Salgueiro~enwiki, Catgut, R'n'B, VolkovBot, Anonymous Dissident, Geometry guy,Forwardmeasure, Soler97, He7d3r, Algebran, Addbot, PV=nRT, Jarble, Yobot, Calle, KamikazeBot, IRP, Drilnoth, Theprogram, J04n,Omnipaedista, Charvest, WaysToEscape, CESSMASTER,Martlet1215, Makki98, RjwilmsiBot, KHamsun, Quondum, D.Lazard, Super-real dance, Movses-bot, MerlIwBot, Deltahedron, CsDix, Monkbot and Anonymous: 50

Alternative algebra Source: https://en.wikipedia.org/wiki/Alternative_algebra?oldid=634813045 Contributors: AxelBoldt, Zundark,Toby Bartels, David spector, Michael Hardy, Silversh, Loren Rosen, Charles Matthews, Fredrik, Robinh, Fropu, Pharotic, Almit39,Klemen Kocjancic, Rich Farmbrough, El C, Rgdboer, AllyUnion, Drbreznjev, Oleg Alexandrov, R.e.b., Sodin, Ihope127, Cesium 133,Cydebot, SMWatt, Konradek, Brian Human, Addbot, Ersik, Joule36e5, Charvest, AdventurousSquirrel, ChrisGualtieri, Deltahedron,SoledadKabocha, JMP EAX and Anonymous: 8

Associative property Source: https://en.wikipedia.org/wiki/Associative_property?oldid=671922176 Contributors: AxelBoldt, Zundark,Jeronimo, Andre Engels, XJaM, Christian List, Toby~enwiki, Toby Bartels, Patrick, Michael Hardy, Ellywa, Andres, Pizza Puzzle,Ideyal, Charles Matthews, Dysprosia, Kwantus, Robbot, Mattblack82, Wikibot, Wereon, Robinh, Tobias Bergemann, Giftlite, Smjg,Lethe, Herbee, Brona, Jason Quinn, Deleting Unnecessary Words, Creidieki, Rich Farmbrough, Guanabot, Paul August, Rgdboer, Jum-buck, Alansohn, Gary, Burn, H2g2bob, Bsadowski1, Oleg Alexandrov, Linas, Palica, Graham87, Deadcorpse, SixWingedSeraph, Fre-plySpang, Yurik, Josh Parris, Rjwilmsi, Salix alba, Mathbot, Chobot, YurikBot, Thane, AdiJapan, BOT-Superzerocool, Bota47, Banus,KnightRider~enwiki, Mmernex, Melchoir, PJTraill, Thumperward, Fuzzform, SchftyThree, Octahedron80, Zven, Furby100, Wine Guy,Smooth O, Cybercobra, Cothrun, Will Beback, SashatoBot, IronGargoyle, Physis, 16@r, JRSpriggs, CBM, Gregbard, Xtv, Mr Gronk,Rbanzai, Thijs!bot, Egrin, Marek69, Escarbot, Salgueiro~enwiki, DAGwyn, David Eppstein, JCraw, Anaxial, Trusilver, Acalamari,Pyrospirit, Krishnachandranvn, Daniel5Ko, GaborLajos, Darklich14, LokiClock, AlnoktaBOT, Rei-bot, Wikiisawesome, SieBot, Ger-akibot, Flyer22, Hello71, Trefgy, Classicalecon, ClueBot, The Thing That Should Not Be, Cli, Razimantv, Mudshark36, Auntof6,DragonBot, Watchduck, Bender2k14, SoxBot III, Addbot, Some jerk on the Internet, Download, BepBot, Tide rolls, Jarble, Luckas-bot,Yobot, TaBOT-zerem, Kuraga, MauritsBot, GrouchoBot, PI314r, Charvest, Ex13, Erik9bot, MacMed, Pinethicket, MastiBot, Fox Wil-son, Ujoimro, GoingBatty, ZroBot, NuclearDuckie, Quondum, D.Lazard, EdoBot, Crown Prince, ClueBot NG, Wcherowi, Kevin Gor-man, Widr, Furkaocean, IkamusumeFan, SuperNerd137, Rang213, Stephan Kulla, Lugia2453, Jshynn, Epicgenius, Kenrambergm2374,Matthew Kastor, Subcientico, Monarchrob1 and Anonymous: 119

Flexible algebra Source: https://en.wikipedia.org/wiki/Flexible_algebra?oldid=633557078 Contributors: Michael Hardy, R.e.b., Cyde-bot, Magioladitis, JohnBlackburne, Addbot, Slow Phil, Quondum and Deltahedron

Non-associative algebra Source: https://en.wikipedia.org/wiki/Non-associative_algebra?oldid=672106189 Contributors: TakuyaMu-rata, Charles Matthews, BD2412, Rjwilmsi, R.e.b., Arthur Rubin, Reyk, PJTraill, Bill Shillito, Rschwieb, Cydebot, Vanish2, David Epp-stein, Robin S, Thehotelambush, Balrivo, He7d3r, DumZiBoT, Addbot, Ayacop, Yobot, AnomieBOT, Charvest, FrescoBot, ZroBot,Quondum, Helpful Pixie Bot, Deltahedron, Lichelder, CsDix, Jose Brox, Cardinal Nimber and Anonymous: 7

Octonion Source: https://en.wikipedia.org/wiki/Octonion?oldid=651438693 Contributors: AxelBoldt, Mav, Zundark, Taw, Mjb, Bde-sham, Michael Hardy, TakuyaMurata, Tristanb, Ideyal, Charles Matthews, Zoicon5, Sabbut, Donarreiskoer, Robinh, Fuelbottle, TobiasBergemann, Tosha, Giftlite, Fropu, Dmmaus, Pmanderson, Icairns, Sam Hocevar, Lumidek, Karl Dickman, Chris Howard, Qutezuce,Zaslav, Oleg Alexandrov, Linas, Jacobolus, Isnow, Graham87, Rjwilmsi, Isaac Rabinovitch, OneWeirdDude, R.e.b., FlaBot, Mathbot,ChongDae, Chobot, RussBot, JoeBruno, Bota47, Ms2ger, Jeppesn, Bluebot, Silly rabbit, Nbarth, Hgrosser, DRLB, Tesseran, Loadmas-ter, Saxbryn, Normmit, MatthewMain, 345Kai, TheTito, Myasuda, Buttonius, Michael C Price, Koeplinger, Pajz, Baccyak4H, DAGwyn,STBot, Iida-yosiaki, Policron, VolkovBot, JohnBlackburne, AlnoktaBOT, Cbigorgne, Nousernamesleft, Antixt, Markherm, AlleborgoBot,Davekaminski, Henry Delforn (old), Alexbot, He7d3r, Bender2k14, Sun Creator, Brews ohare, Addbot, CarsracBot, Lightbot, PV=nRT,Legobot, Luckas-bot, Yobot, Jgmoxness, Janet Davis, NobelBot, Shadowjams, Howard McCay, Aleks kleyn, Machine Elf 1735, Citationbot 1, Heptadecagon, RedBot, Casimir9999, , Unbitwise, RjwilmsiBot, Afteread, EmausBot, Bor75, ZroBot, Quondum,Nemo47, Ebehn, Anita5192, Lanthanum-138, Frietjes, Helpful Pixie Bot, BG19bot, Dylanvt, Deltahedron, OrthogonalFrog, Hladky77,BethNaught, Scottrileywilson and Anonymous: 63

6.8.2 Images File:Associativity_of_real_number_addition.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f6/Associativity_of_real_

number_addition.svg License: CC BY 3.0 Contributors: Own work Original artist: Stephan Kulla (User:Stephan Kulla) File:FanoPlane.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/2d/FanoPlane.svg License: CC BY-SA 3.0 Contribu-

tors: Own work Original artist: Jgmoxness File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0

Contributors:Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007

File:Semigroup_associative.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/80/Semigroup_associative.svg License:CC BY-SA 4.0 Contributors: Own work Original artist: IkamusumeFan

6.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 33

File:Tamari_lattice.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/46/Tamari_lattice.svgLicense: Public domainCon-tributors: Own work Original artist: David Eppstein

File:Text_document_with_red_question_mark.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg License: Public domain Contributors: Created by bdesham with Inkscape; based upon Text-x-generic.svgfrom the Tango project. Original artist: Benjamin D. Esham (bdesham)

File:Wiktionary-logo-en.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f8/Wiktionary-logo-en.svg License: Publicdomain Contributors: Vector version of Image:Wiktionary-logo-en.png. Original artist: Vectorized by Fvasconcellos (talk contribs),based on original logo tossed together by Brion Vibber

6.8.3 Content license Creative Commons Attribution-Share Alike 3.0

Algebra over a fieldDefinition and motivation First example: The complex numbers Definition A motivating example: quaternions Another motivating example: the cross product

Basic concepts Algebra homomorphisms Subalgebras and ideals Extension of scalars

Kinds of algebras and examples Unital algebra Zero algebra Associative algebra Non-associative algebra

Algebras and ringsStructure coefficients Classification of low-dimensional algebrasSee also NotesReferences

Alternative algebraThe associatorExamplesPropertiesApplicationsSee alsoReferencesExternal links

Associative propertyDefinition Generalized associative lawExamplesPropositional logic Rule of replacement Truth functional connectives

Non-associativity Nonassociativity of floating point calculationNotation for non-associative operations

See alsoReferences

Flexible algebraExamplesSee also References

Non-associative algebraAlgebras satisfying identities Associator

Examples Free non-associative algebraAssociated algebrasDerivation algebraEnveloping algebra

See also Notes References

OctonionDefinition CayleyDickson constructionFano plane mnemonicConjugate, norm, and inverse

Properties Commutator and cross productAutomorphismsIsotopies

Integral octonionsSee alsoNotesReferencesExternal linksText and image sources, contributors, and licensesTextImagesContent license