algebra over a field

Algebra over a fieldFrom Wikipedia, the free encyclopedia

Contents

1 Algebra over a field 11.1 Definition and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 First example: The complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Classification 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 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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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 floating point calculation . . . . . . . . . . . . . . . . . . . . . . . . . 163.5.2 Notation for non-associative operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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

4 Non-associative algebra 194.1 Algebras satisfying identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.1 Associator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Free non-associative algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Associated algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.4.1 Derivation algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4.2 Enveloping algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Chapter 1

Algebra over a field

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 field is a vector space equipped with a bilinear product. Thus, an algebra overa field is a set, together with operations of multiplication, addition, and scalar multiplication, by elements of theunderlying field, 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 field 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 field of scalars by a commutative ring, and thus defining 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 Definition 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 field 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 define: (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 fits into the following definition by taking the field K to be the real numbers, and the vector space A tobe the complex numbers.

1
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2 CHAPTER 1. ALGEBRA OVER A FIELD

1.1.2 Definition

Let K be a field, 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 field 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 field 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 defined 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 field.

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
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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 differonly by notation.

1.2.2 Subalgebras and ideals

Main article: Substructure

A subalgebra of an algebra over a field 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 define 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 definition is different from the definition 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 field extension F/K, which is to say a bigger field 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 biggerfield, 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 examples

Algebras over fields come in many different types. These types are specified by insisting on some further axioms,such as commutativity or associativity of the multiplication operation, which are not required in the broad definitionof an algebra. The theories corresponding to the different types of algebras are often very different.

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.
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1.3.2 Zero algebra

An 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 define a unital zero algebra by taking the direct sum of modules of a field (or more generally a ring) Kand a K-vector space (or module) V, and defining 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 field. 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 modification, any algorithmand any software for computing Grbner bases of ideals.

1.3.3 Associative algebra

Main article: Associative algebra

the algebra of all n-by-n matrices over the field (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 defined on the interval [0,1],or the C-algebra of all holomorphic functions defined on some fixed 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 defined 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 algebra

Main article: Non-associative algebra

A non-associative algebra[3] (or distributive algebra) over a field 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
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1.4. ALGEBRAS AND RINGS 5

Alternative algebras

Flexible algebras

Power-associative algebras

1.4 Algebras and rings

The definition of an associative K-algebra with unit is also frequently given in an alternative way. In this case, analgebra over a field 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 definition 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 defined by , which one may write as

f(ka) = kf(a)

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

KA B

Af

B

1.5 Structure coefficients

For algebras over a field, 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 satisfiesthe algebra laws.Thus, given the field K, any finite-dimensional algebra can be specified up to isomorphism by giving its dimension(say n), and specifying n3 structure coefficients ci,j,k, which are scalars. These structure coefficients determine themultiplication in A via the following rule:

eiej =n

k=1

ci,j,kek

where e1,...,en form a basis of A.Note however that several different sets of structure coefficients can give rise to isomorphic algebras.
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When the algebra can be endowed with a metric, then the structure coefficients are written with upper and lowerindices, so as to distinguish their transformation properties under coordinate transformations. Specifically, lowerindices are covariant indices, and transform via pullbacks, while upper indices are contravariant, transforming underpushforwards. Thus, in mathematical physics, the structure coefficients are often written ci,jk, and their defining 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 field, 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 specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up toisomorphism.

1.6 Classification of low-dimensional algebras

Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex num-bers were completely classified up to isomorphism by Eduard Study.[4]

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

1 1 = 1 , 1 a = a , a 1 = a .

It remains to specify

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

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

aa = a , bb = b , ab = ba = 0 for the first 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 fifth algebra.

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

1.7 See also Clifford algebra

Differential algebra

Geometric algebra

Max-plus algebra

Zariskis lemma

Mutation (algebra)
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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
https://books.google.com/books?id=AibpdVNkFDYC&pg=PA3&dq=%2522an+algebra+over+a+field+k%2522https://en.wikipedia.org/wiki/Special:BookSources/0486688135http://www.gutenberg.org/ebooks/25156https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252FBF01692479https://en.wikipedia.org/wiki/Michiel_Hazewinkelhttps://en.wikipedia.org/wiki/Special:BookSources/1402026900

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 associator

Alternative 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 definition 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 flexible 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 satisfies any two of:

8
https://en.wikipedia.org/wiki/Abstract_algebrahttps://en.wikipedia.org/wiki/Algebra_over_a_fieldhttps://en.wikipedia.org/wiki/Associativehttps://en.wikipedia.org/wiki/Alternativityhttps://en.wikipedia.org/wiki/Associative_algebrahttps://en.wikipedia.org/wiki/Non-associative_algebrahttps://en.wikipedia.org/wiki/Octonionhttps://en.wikipedia.org/wiki/Sedenionhttps://en.wikipedia.org/wiki/Associatorhttps://en.wikipedia.org/wiki/Alternating_formhttps://en.wikipedia.org/wiki/Multilinear_maphttps://en.wikipedia.org/wiki/Skew-symmetric_graphhttps://en.wikipedia.org/wiki/Permutationhttps://en.wikipedia.org/wiki/Flexible_identity

2.2. EXAMPLES 9

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

is alternative and therefore satisfies all three identities.An alternating associator is always totally skew-symmetric. The converse holds so long as the characteristic of thebase field 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 Properties

Artins 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 finite-dimensional non-associative alternative algebra is a generalised octonion algebra.[6]

2.4 Applications

The 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 field of A, and A is alternative, then (A,n) is a composition algebra.
https://en.wikipedia.org/wiki/Characteristic_(algebra)https://en.wikipedia.org/wiki/Associative_algebrahttps://en.wikipedia.org/wiki/Octonionhttps://en.wikipedia.org/wiki/Octonion_algebrahttps://en.wikipedia.org/wiki/Subalgebrahttps://en.wikipedia.org/wiki/Associativehttps://en.wikipedia.org/wiki/Power-associativehttps://en.wikipedia.org/wiki/Sedenionhttps://en.wikipedia.org/wiki/Moufang_identitieshttps://en.wikipedia.org/wiki/Moufang_loophttps://en.wikipedia.org/wiki/Group_of_unitshttps://en.wikipedia.org/wiki/Group_of_unitshttps://en.wikipedia.org/wiki/Moufang_planehttps://en.wikipedia.org/wiki/Composition_algebrahttps://en.wikipedia.org/wiki/Involution_(mathematics)https://en.wikipedia.org/wiki/Anti-automorphismhttps://en.wikipedia.org/wiki/Linear_span

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
https://en.wikipedia.org/wiki/Zorn_ringhttps://en.wikipedia.org/wiki/Maltsev_algebrahttps://en.wikipedia.org/wiki/John_Horton_Conwayhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/1-56881-134-9https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:1098.17001https://en.wikipedia.org/wiki/American_Mathematical_Societyhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-486-68813-5http://www.encyclopediaofmath.org/index.php?title=Alternative_rings_and_algebrashttps://en.wikipedia.org/wiki/Encyclopedia_of_Mathematicshttps://en.wikipedia.org/wiki/Encyclopedia_of_Mathematicshttps://en.wikipedia.org/wiki/Springer_Science+Business_Mediahttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/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 floating point numbers in com-puter science is not associative, and is an important source of rounding error.

3.1 Definition

Formally, a binary operation on a set S is called associative if it satisfies 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
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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 law

If 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 five 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 Examples

Some examples of associative operations include the following.

The concatenation of the three strings hello, " ", world can be computed by concatenating the first twostrings (giving hello ") and appending the third string (world), or by joining the second and third string(giving " world) and concatenating the first 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.,
https://en.wikipedia.org/wiki/Commutative_diagramhttps://en.wikipedia.org/wiki/Function_compositionhttps://en.wikipedia.org/wiki/Catalan_number#Applications_in_combinatoricshttps://en.wikipedia.org/wiki/String_concatenationhttps://en.wikipedia.org/wiki/Arithmetichttps://en.wikipedia.org/wiki/Additionhttps://en.wikipedia.org/wiki/Multiplicationhttps://en.wikipedia.org/wiki/Real_number

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, five factors a, b, c, d, e result in a Tamari lattice of order four, possibly different products.

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

}for all x, y, z R.

Because of associativity, the grouping parentheses can be omitted without ambiguity.
https://en.wikipedia.org/wiki/Tamari_lattice


(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 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 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.
https://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Quaternionhttps://en.wikipedia.org/wiki/Octonionhttps://en.wikipedia.org/wiki/Greatest_common_divisorhttps://en.wikipedia.org/wiki/Least_common_multiplehttps://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Union_(set_theory)https://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Functional_compositionhttps://en.wikipedia.org/wiki/Matrix_(mathematics)https://en.wikipedia.org/wiki/Linear_maphttps://en.wikipedia.org/wiki/Matrix_multiplication

3.4. PROPOSITIONAL LOGIC 15

3.4 Propositional logic

3.4.1 Rule of replacement

In 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 connectives

Associativity 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-associativity

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

(x y) z = x (y z) for some x, y, z S.

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

Subtraction

(5 3) 2 = 5 (3 2)
https://en.wikipedia.org/wiki/Validityhttps://en.wikipedia.org/wiki/Rule_of_replacementhttps://en.wikipedia.org/wiki/Well-formed_formulahttps://en.wikipedia.org/wiki/Formal_proofhttps://en.wikipedia.org/wiki/Metalogichttps://en.wikipedia.org/wiki/Symbol_(formal)https://en.wikipedia.org/wiki/Formal_proofhttps://en.wikipedia.org/wiki/Logical_connectivehttps://en.wikipedia.org/wiki/Propositional_logichttps://en.wikipedia.org/wiki/Logical_equivalencehttps://en.wikipedia.org/wiki/Logical_equivalencehttps://en.wikipedia.org/wiki/Tautology_(logic)https://en.wikipedia.org/wiki/Subtraction


Division

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

Exponentiation

2(12) = (21)2

Also note that infinite 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 different 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 infinitesimal transformations, andhave become ubiquitous in mathematics.There are other specific types of non-associative structures that have been studied in depth; these tend to come fromsome specific applications or areas such as combinatorial mathematics. Other examples are Quasigroup, Quasifield,Non-associative ring, Non-associative algebra and Commutative non-associative magmas.

3.5.1 Nonassociativity of floating point calculation

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

To illustrate this, consider a floating 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.

for all w, x, y, z Swhile a right-associative operation is conventionally evaluated from right to left:
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3.5. NON-ASSOCIATIVITY 17

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

for all w, x, y, z 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 R;x/y/z = (x/y)/z for all x, y, z R with y = 0, z = 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 definition

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 defined include the following.

Taking the Cross product of three vectors:

a (b c) = (a b) c for some a, b, c R3

Taking the pairwise average of real numbers:

(x+ y)/2 + z

2= x+ (y + z)/2

2for all x, y, z R with x = z.

Taking the relative complement of sets (A\B)\C is not the same as A\(B\C) . (Compare material nonim-plication in logic.)
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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. Definition 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, Effects 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
https://en.wikipedia.org/wiki/Light%2527s_associativity_testhttps://en.wikipedia.org/wiki/Semigrouphttps://en.wikipedia.org/wiki/Commutativityhttps://en.wikipedia.org/wiki/Distributivityhttps://en.wikipedia.org/wiki/Power_associativityhttps://en.wikipedia.org/wiki/Alternativityhttps://en.wikipedia.org/wiki/N-ary_associativityhttps://en.wikipedia.org/wiki/Springer_Science+Business_Mediahttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0387905189http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000258.htmlhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-471-51001-7https://en.wikipedia.org/wiki/The_Art_of_Computer_Programminghttp://ieeexplore.ieee.org/servlet/opac?punumber=4610933https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1109%252FIEEESTD.2008.4610935https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-7381-5753-5http://cass-mt.pnnl.gov/docs/pubs/pnnleffects_of_floating-pointpaper.pdfhttp://cass-mt.pnnl.gov/docs/pubs/pnnleffects_of_floating-pointpaper.pdfhttp://cass-mt.pnnl.gov/docs/pubs/pnnleffects_of_floating-pointpaper.pdfhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1145%252F103162.103163

Chapter 4

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 field (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 differentanswers.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".

4.1 Algebras satisfying identities

Ring-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).

19
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20 CHAPTER 4. NON-ASSOCIATIVE ALGEBRA

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 flexible.[14][15]

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

4.1.1 Associator

Main 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 identitiessatisfied 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] = {0} .

4.2 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 satisfies the Jacobi identity.

Lie algebras are algebras satisfying anticommutativity and the Jacobi identity.

Algebras of vector fields on a differentiable 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 field of characteristic other than 2 gives rise to a Jordan algebra by defining 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 fields. Allassociative algebras are alternative. Up to isomorphism, the only finite-dimensional real alternative, divisionalgebras (see below) are the reals, complexes, quaternions and octonions.
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4.3. FREE NON-ASSOCIATIVE ALGEBRA 21

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 filtered algebras.

Division algebras, in which multiplicative inverses exist. The finite-dimensional alternative division algebrasover the field of real numbers have been classified. 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 field, and e a unitfor the algebra. Examples include all finite-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 different ways.

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

4.3 Free non-associative algebra

The free non-associative algebra on a set X over a field K is defined as the algebra with basis consisting of all non-associative monomials, finite 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]

4.4 Associated algebras

An algebra A over a field 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.

4.4.1 Derivation algebra

A 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]
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4.4.2 Enveloping algebra

There 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 satisfied 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 defined by[25]

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

or equivalently

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

4.5 See also List of algebras

Commutative non-associative magmas, which give rise to non-associative algebras

4.6 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 fields 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
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4.7. REFERENCES 23

[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

4.7 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
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https://en.wikipedia.org/wiki/Algebra_over_a_field?oldid=669778616https://en.wikipedia.org/wiki/Alternative_algebra?oldid=634813045https://en.wikipedia.org/wiki/Associative_property?oldid=671922176https://en.wikipedia.org/wiki/Non-associative_algebra?oldid=672106189https://upload.wikimedia.org/wikipedia/commons/f/f6/Associativity_of_real_number_addition.svghttps://upload.wikimedia.org/wikipedia/commons/f/f6/Associativity_of_real_number_addition.svghttp://kulla.me///commons.wikimedia.org/wiki/User:Stephan_Kullahttps://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg//en.wikipedia.org/wiki/File:Question_book.png//en.wikipedia.org/wiki/User:Equazcion//en.wikipedia.org/wiki/User:Tkgd2007https://upload.wikimedia.org/wikipedia/commons/8/80/Semigroup_associative.svg//commons.wikimedia.org/wiki/User:IkamusumeFanhttps://upload.wikimedia.org/wikipedia/commons/4/46/Tamari_lattice.svg//commons.wikimedia.org/wiki/User:David_Eppsteinhttps://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svghttps://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg//commons.wikimedia.org/wiki/User:Bdesham//commons.wikimedia.org/wiki/File:Text-x-generic.svg//commons.wikimedia.org/wiki/User:Bdeshamhttps://upload.wikimedia.org/wikipedia/commons/f/f8/Wiktionary-logo-en.svg//commons.wikimedia.org/wiki/File:Wiktionary-logo-en.png//commons.wikimedia.org/wiki/User:Fvasconcellos//commons.wikimedia.org/wiki/User_talk:Fvasconcellos//commons.wikimedia.org/wiki/Special:Contributions/Fvasconcellos//commons.wikimedia.org/wiki/User:Brion_VIBBERhttps://creativecommons.org/licenses/by-sa/3.0/Algebra over a fieldDefinition and motivation First example: The complex numbers Definition A motivating example: quaternions Another motivating example: the cross productBasic 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 NotesReferencesAlternative algebraThe associatorExamplesPropertiesApplicationsSee alsoReferencesExternal linksAssociative propertyDefinition Generalized associative lawExamplesPropositional logic Rule of replacement Truth functional connectives Non-associativity Nonassociativity of floating point calculationNotation for non-associative operations See alsoReferencesNon-associative algebraAlgebras satisfying identities AssociatorExamples Free non-associative algebraAssociated algebrasDerivation algebraEnveloping algebraSee also Notes References Text and image sources, contributors, and licensesTextImagesContent license

algebra over a field

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