Alternative algebra: Difference between revisions
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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]] | |||
Revision as of 05:39, 14 June 2013
Template:More footnotes In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
for all x and y in the algebra. Every associative algebra is obviously alternative, but so too are some strictly nonassociative algebras such as the octonions. The sedenions, 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. The associator is a trilinear map given by
![{\displaystyle [x,y,z]=(xy)z-x(yz)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef1dd4cbc4e3e9a80e2ec6866d1294ef04efe97)
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[1]
![{\displaystyle [x,x,y]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a1374960ee08cdee5d041c68cab56a4997b5ea)
![{\displaystyle [y,x,x]=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/beeab85c1f13c8a8fd1653b39fd5bd6b718306ec)
Both of these identities together imply that the associator is totally skew-symmetric. That is,
![{\displaystyle [x_{\sigma (1)},x_{\sigma (2)},x_{\sigma (3)}]=\operatorname {sgn}(\sigma )[x_{1},x_{2},x_{3}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bae2b45897eaef18c2bc50d102fc7f2888ab884)
for any permutation σ. It follows that
![{\displaystyle [x,y,x]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4524ce3f04a28c8e5bc542173a6223f00b0412f)
for all x and y. This is equivalent to the flexible identity[2]
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:
- right alternative identity:
- flexible identity:
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 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.[3] 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 in an alternative algebra associate (i.e. ) the subalgebra generated by those elements is associative.
![{\displaystyle [x,y,z]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95cef9c8a4542f9ded3ac917172894e992a8f533)
A corollary of Artin's theorem is that alternative algebras are power-associative, that is, the subalgebra generated by a single element is associative.[4] The converse need not hold: the sedenions are power-associative but not alternative.
hold in any alternative algebra.[2]
In a unital alternative algebra, multiplicative inverses are unique whenever they exist. Moreover, for any invertible element and all one has
This is equivalent to saying the associator vanishes for all such and . If and are invertible then is also invertible with inverse . 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.
![{\displaystyle [x^{-1},x,y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b240bc5bf9c148e84b2502f1442c3931f5654e4)
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 such that a + 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-singular mapping into the field of A, and A is alternative, then (A,n) is a composition algebra.
See also
References
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- 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.
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