- Alternative algebra
In

abstract algebra , an**alternative algebra**is an algebra in which multiplication need not beassociative , only alternative. That is, one must have

*$x(xy)\; =\; (xx)y$

*$(yx)x\; =\; y(xx)$for all "x" and "y" in the algebra. Everyassociative algebra is obviously alternative, but so too are some strictlynonassociative algebra s such as theoctonion s. Thesedenion 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. 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. 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 totallyskew-symmetric . That is,:$[x\_\{sigma(1)\},\; x\_\{sigma(2)\},\; x\_\{sigma(3)\}]\; =\; sgn(sigma)\; [x\_1,x\_2,x\_3]$for anypermutation σ. It follows that:$[x,y,x]\; =\; 0$for all "x" and "y". This is equivalent to the so-calledflexible 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.

**Properties****Artin's theorem**states that in an alternative algebra thesubalgebra generated by any two elements isassociative . 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. The converse need not hold: thesedenion s 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.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 thegroup of units in an associative ring or algebra.**References***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

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