In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Mathematics, associativity is a property that a Binary operation can have In Abstract algebra, a magma G is said to be left alternative if ( xx) y = x ( xy) for all x and y 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 nonassociative algebras such as the octonions. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In Mathematics, non-associative algebra is a subfield of Abstract algebra, in which are studied Algebraic structures endowed with a Binary operation In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real The sedenions, on the other hand, are not alternative. In Abstract algebra, sedenions form a 16- dimensional algebra over the reals.
The associator
Alternative algebras are so named because they are precisely the algebras for which the associator is alternating. In Abstract algebra, for a ring or algebra R the associator is the Multilinear map R \times R \times R \to R given 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 it arguments are equal. In Linear algebra, a multilinear map is a Mathematical function of several vector variables that is linear in each variable The left and right alternative identities for an algebra are equivalent to
- [x,x,y] = 0
- [y,x,x] = 0.
Both of these identities together imply that the associator is totally skew-symmetric. That is,
- [xσ(1),xσ(2),xσ(3)] = sgn(σ)[x1,x2,x3]
for any permutation σ. In several fields of Mathematics the term permutation is used with different but closely related meanings It follows that
- [x,y,x] = 0
for all x and y. This is equivalent to the so-called flexible identity
- (xy)x = x(yx).
The associator 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. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's
Properties
Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative. In Algebra (mathematics, the word "algebra" usually means a Vector space or module equipped with an additional bilinear operation In Mathematics, associativity is a property that a Binary operation can have 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. In Abstract algebra, power associativity is a weak form of Associativity. The converse need not hold: the sedenions are power-associative but not alternative. In Abstract algebra, sedenions form a 16- dimensional algebra over the reals.
- a(x(ay)) = (axa)y
- ((xa)y)a = x(aya)
- (ax)(ya) = a(xy)a
hold in any alternative algebra. In Mathematics, a Moufang loop is a special kind of Algebraic structure.
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 − 1x − 1. The set of all invertible elements is therefore closed under multiplication and forms a Moufang loop. In Mathematics, a Moufang loop is a special kind of Algebraic structure. This loop of units in an alternative ring or algebra is analogous to the group of units in an associative ring or algebra. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i
References
- Schafer, Richard D. (1995). An Introduction to Nonassociative Algebras. New York: Dover Publications. ISBN 0-486-68813-5.
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