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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 [[Algebra over a field#Non-associative algebras|nonassociative algebras]] such as the [[octonion]]s. The [[sedenion]]s, on the other hand, are not alternative.
 
==The associator==
 
Alternative algebras are so named because they are precisely the algebras for which the [[associator]] is [[alternating form|alternating]]. The associator is a [[multilinear map|trilinear map]] given by
:<math>[x,y,z] = (xy)z - x(yz)</math>
By definition a multilinear map is alternating if it 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 the associator is 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]] σ. It follows that
:<math>[x,y,x] = 0</math>
for all ''x'' and ''y''. 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. 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. The converse holds so long as the [[characteristic (algebra)|characteristic]] of the base field is not 2.
 
==Properties==
'''Artin's theorem''' states that in an alternative algebra the [[subalgebra]] generated by any two elements is [[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 without parenthesis unambiguously 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 [[sedenion]]s 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 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.
 
==Applications==
The projective plane over any alternative division ring is a [[Moufang plane]].
 
The close relationship of alternative algebras and [[composition algebra]]s was given by Guy Roos in 2008: He shows (page 162) the relation for an algebra ''A'' with unit element ''e'' and an [[involution (mathematics)|involutive]] [[anti-automorphism]] <math>a \mapsto a^*</math> such that ''a'' + ''a''* and ''aa''* are on the line [[linear span|spanned]] by ''e'' for all ''a'' in ''A''. Use the notation ''n''(''a'') = ''aa''*. Then if ''n'' is a non-singular mapping into the field of ''A'', and ''A'' is alternative, then (''A,n'') is a composition algebra.
 
==See also==
 
*[[Zorn ring]]
* [[Maltsev algebra]]
 
==References==
{{reflist}}
* 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]].
*{{Cite book | first = Richard D. | last = Schafer | title = An Introduction to Nonassociative Algebras | publisher = Dover Publications | location = New York | year = 1995 | isbn = 0-486-68813-5}}
 
==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]]
 
[[fr:Alternativité]]

Latest revision as of 11:53, 21 November 2014

Gourmet coffee roasters are a necessity whenever roasting coffee beans. The fresh smells of beans being heated and the aromas wafting through the air will make any coffee-lovers heart go pitter-patter. These fantastic inventions are basically gourmet machines which are prepared to roast coffee beans to the desired level of roasting. What level of roasting is about we and what a palate prefers!

The green coffee weight loss extract technique of losing weight is one superior option to get rid of those extra fats. One might ask why these beans are green. It is considering these beans have not been roasted. The reason why the regular coffee beans many individuals are familiar with are dark inside color is because these beans have been roasted as much as 475 levels Fahrenheit. This roasting process can really make the beans lose its fat-burning and anti-oxidant element which it naturally possess. Needless to say, the green coffee weight losss are in its many natural state plus therefore will make individuals lose fat naturally.

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The largest community of scientists has completed a study on the advantages of using green coffee extract for fat reduction. They gave an extract of green coffee to a sizable group of individuals. They told to the group to not change anything about their lifestyle. This is how they were able to isolate the benefits of the green coffee extract.

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