Alternative algebra

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In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. 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 the octonions. The sedenions, on the other hand, are not alternative.

The associator[edit]

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

[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 and right 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_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}] = \sgn(\sigma)[x_1,x_2,x_3]

for any permutation σ. 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 alternating is clearly alternative. By symmetry, any algebra which satisfies any two of:

  • 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 the base field is not 2.

Properties[edit]

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 x,y,z in an alternative algebra associate (i.e. [x,y,z] = 0) 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.[4] 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 invertible element x and all y one has

y = x^{-1}(xy).

This is equivalent to saying the associator [x^{-1},x,y] vanishes for all such x and y. If x and y are invertible then xy is also invertible with inverse (xy)^{-1} = y^{-1}x^{-1}. 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[edit]

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 \mapsto a^* 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[edit]

References[edit]

  1. ^ Schafer (1995) p.27
  2. ^ a b Schafer (1995) p.28
  3. ^ Schafer (1995) p.29
  4. ^ Schafer (1995) p.30
  • 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.
  • Schafer, Richard D. (1995). An Introduction to Nonassociative Algebras. New York: Dover Publications. ISBN 0-486-68813-5. 

External links[edit]