Editing Alternative algebra

{{Short description|1=Algebra where x(xy)=(xx)y and (yx)x=y(xx)}}

In [[abstract algebra]], an '''alternative algebra''' is an [[algebra over a field|algebra]] in which multiplication need not be [[associative]], only [[alternativity|alternative]]. That is, one must have
*<math>x(xy) = (xx)y</math>
*<math>(yx)x = y(xx)</math>
for all ''x'' and ''y'' in the algebra.

Every [[associative algebra]] is obviously alternative, but so too are some strictly [[non-associative algebra]]s such as the [[octonion]]s.

==The associator==

Alternative algebras are so named because they are the algebras for which the [[associator]] is [[alternating form|alternating]]. The associator is a [[trilinear map]] given by
:<math>[x,y,z] = (xy)z - x(yz)</math>.
By definition, a [[multilinear map]] is alternating if it [[Vanish_(mathematics)|vanishes]] whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to<ref name=Sch27>Schafer (1995) p. 27</ref>
:<math>[x,x,y] = 0</math>
:<math>[y,x,x] = 0</math>
Both of these identities together imply that:
:<math>[x,y,x]=[x,x,x]+[x,y,x]+</math>
:<math>-[x,x+y,x+y] =</math>
:<math>= [x,x+y,-y] =</math>
:<math>= [x,x,-y] - [x,y,y] = 0</math>
for all <math>x</math> and <math>y</math>. This is equivalent to the ''[[flexible identity]]''<ref name=Sch28>Schafer (1995) p. 28</ref>
:<math>(xy)x = x(yx).</math>
The associator of an alternative algebra is therefore alternating. [[Converse (logic)|Conversely]], any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of:
*left alternative identity: <math>x(xy) = (xx)y</math>
*right alternative identity: <math>(yx)x = y(xx)</math>
*flexible identity: <math>(xy)x = x(yx).</math>
is alternative and therefore satisfies all three identities.

An alternating associator is always totally skew-symmetric. That is,
:<math>[x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}] = \sgn(\sigma)[x_1,x_2,x_3]</math>
for any [[permutation]] <math>\sigma</math>. The converse holds so long as the [[characteristic (algebra)|characteristic]] of the base [[field (mathematics)|field]] is not 2.

==Examples==
* Every associative algebra is alternative.
* The [[octonion]]s form a non-associative alternative algebra, a [[normed division algebra]] of dimension 8 over the [[real number]]s.<ref>{{cite book | author-link=John Horton Conway | last1=Conway | first1=John Horton | last2=Smith | first2=Derek A. | year=2003 | title=On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry | publisher=A. K. Peters | isbn=1-56881-134-9 | zbl=1098.17001 }}</ref>
* More generally, any [[octonion algebra]] is alternative.

===Non-examples===
* The [[sedenion]]s, [[trigintaduonion]]s, and all higher [[Cayley–Dickson algebra]]s lose alternativity.

==Properties==
{{Redirect|Artin's theorem|Artin's theorem on primitive elements|Primitive element theorem}}
'''Artin's theorem''' states that in an alternative algebra the [[subalgebra]] generated by any two elements is [[associative algebra|associative]].<ref name=Sch29>Schafer (1995) p. 29</ref>  Conversely, any algebra for which this is true is clearly alternative. It follows that expressions involving only two variables can be written unambiguously without parentheses in an alternative algebra. A generalization of Artin's theorem states that whenever three elements <math>x,y,z</math> in an alternative algebra associate (i.e., <math>[x,y,z] = 0</math>), the subalgebra generated by those elements is associative.

A [[corollary]] of Artin's theorem is that alternative algebras are [[power-associative]], that is, the subalgebra generated by a single element is associative.<ref name=Sch30>Schafer (1995) p. 30</ref>  The converse need not hold: the sedenions are power-associative but not alternative.

The [[Moufang identities]]
*<math>a(x(ay)) = (axa)y</math>
*<math>((xa)y)a = x(aya)</math>
*<math>(ax)(ya) = a(xy)a</math>
hold in any alternative algebra.<ref name=Sch28/>

In a unital alternative algebra, multiplicative [[inverse element|inverses]] are unique whenever they exist. Moreover, for any invertible element <math>x</math> and all <math>y</math> one has
:<math>y = x^{-1}(xy).</math>
This is equivalent to saying the associator <math>[x^{-1},x,y]</math> vanishes for all such <math>x</math> and <math>y</math>. 

If <math>x</math> and <math>y</math> are invertible then <math>xy</math> is also invertible with inverse <math>(xy)^{-1} = y^{-1}x^{-1}</math>. The set of all invertible elements is therefore closed under multiplication and forms a [[Moufang loop]]. This ''loop of units'' in an alternative ring or algebra is analogous to the [[group of units]] in an [[associative ring]] or algebra.

Kleinfeld's theorem states that any simple non-associative alternative ring is a generalized octonion algebra over its [[center (ring theory)|center]].<ref name=ZSSS151>Zhevlakov, Slin'ko, Shestakov, Shirshov. (1982) p. 151</ref>
The structure theory of alternative rings is presented in the book ''Rings That Are Nearly Associative'' by Zhevlakov, Slin'ko, Shestakov, and Shirshov.<ref name=ZSSS>Zhevlakov, Slin'ko, Shestakov, Shirshov (1982)</ref>

==Occurrence==
The [[projective plane]] over any alternative [[division ring]] is a [[Moufang plane]].

Every [[composition algebra]] is an alternative algebra, as shown by Guy Roos in 2008:<ref>Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in ''Symmetries in Complex Analysis'' by Bruce Gilligan & Guy Roos, volume 468 of ''Contemporary Mathematics'', [[American Mathematical Society]]</ref> A composition algebra ''A'' over a field ''K'' has a ''norm n'' that is a multiplicative [[homomorphism]]: <math>n(a \times b) = n(a) \times n(b)</math> connecting (''A'', ×) and (''K'', ×).

Define the form ( _ : _ ): ''A'' × ''A'' → ''K'' by <math>(a:b) = n(a+b) - n(a) - n(b).</math> Then the trace of ''a'' is given by (''a'':1) and the conjugate by ''a''* = (''a'':1)e – ''a'' where e is the basis element for 1. A series of exercises prove that a composition algebra is always an alternative algebra.<ref>{{wikibooks-inline|Associative Composition Algebra/Transcendental paradigm#Categorical treatment}}</ref>

== See also ==

* [[Algebra over a field]]
* [[Maltsev algebra]]
* [[Zorn ring]]

==References==
{{reflist}}
*{{Cite book | first = Richard D. | last = Schafer | author-link = Richard D. Schafer|title = An Introduction to Nonassociative Algebras | publisher = Dover Publications | location = New York | year = 1995 | isbn = 0-486-68813-5 | url = https://archive.org/details/introductiontono0000scha | zbl = 0145.25601 | url-access = registration }}
* {{cite book | first1=K.A. | last1=Zhevlakov | first2=A.M.|last2= Slin'ko | first3= I.P. | last3= Shestakov |first4 =A.I. | last4= Shirshov |year=1982 | orig-year=1978 | zbl=0487.17001 |mr = 0518614 | title=Rings That Are Nearly Associative | publisher=[[Academic Press]] | isbn=0-12-779850-1 }}

==External links==
*{{SpringerEOM|id=Alternative_rings_and_algebras|first=K.A.|last= Zhevlakov|title=Alternative rings and algebras}}

{{DEFAULTSORT:Alternative Algebra}}
[[Category:Non-associative algebras]]