Alternative algebra

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have

• ${\displaystyle x(xy)=(xx)y}$
• ${\displaystyle (yx)x=y(xx)}$

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 associator

Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by

${\displaystyle [x,y,z]=(xy)z-x(yz)}$.

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]

${\displaystyle [x,x,y]=0}$

${\displaystyle [y,x,x]=0.}$

Both of these identities together imply that

${\displaystyle [x,y,x]=[x,x,x]+[x,y,x]-[x,x+y,x+y]=[x,x+y,-y]=[x,x,-y]-[x,y,y]=0}$

for all ${\displaystyle x}$ and ${\displaystyle y}$. This is equivalent to the flexible identity ^[2]

${\displaystyle (xy)x=x(yx).}$

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: ${\displaystyle x(xy)=(xx)y}$
• right alternative identity: ${\displaystyle (yx)x=y(xx)}$
• flexible identity: ${\displaystyle (xy)x=x(yx).}$

is alternative and therefore satisfies all three identities.

An alternating associator is always totally skew-symmetric. That is,

${\displaystyle [x_{\sigma (1)},x_{\sigma (2)},x_{\sigma (3)}]=\operatorname {sgn}(\sigma )[x_{1},x_{2},x_{3}]}$

for any permutation ${\displaystyle \sigma }$. The converse holds so long as the characteristic of the base field is not 2.

Examples

• Every associative algebra is alternative.
• The octonions form a non-associative alternative algebra, a normed division algebra of dimension 8 over the real numbers. ^[3]
• More generally, any octonion algebra is alternative.

Non-examples

• The sedenions and all higher Cayley–Dickson algebras lose alternativity.

Properties

"Artin's theorem" redirects here. For Artin's theorem on primitive elements, see Primitive element theorem.

Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative. ^[4] Conversely, any algebra for which this is true is clearly alternative. It follows that expressions involving only two variables can be written unambiguously without parentheses in an alternative algebra. A generalization of Artin's theorem states that whenever three elements ${\displaystyle x,y,z}$ in an alternative algebra associate (i.e., ${\displaystyle [x,y,z]=0}$), 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. ^[5] The converse need not hold: the sedenions are power-associative but not alternative.

The Moufang identities

• ${\displaystyle a(x(ay))=(axa)y}$
• ${\displaystyle ((xa)y)a=x(aya)}$
• ${\displaystyle (ax)(ya)=a(xy)a}$

hold in any alternative algebra. ^[2]

In a unital alternative algebra, multiplicative inverses are unique whenever they exist. Moreover, for any invertible element ${\displaystyle x}$ and all ${\displaystyle y}$ one has

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

This is equivalent to saying the associator ${\displaystyle [x^{-1},x,y]}$ vanishes for all such ${\displaystyle x}$ and ${\displaystyle y}$.

If ${\displaystyle x}$ and ${\displaystyle y}$ are invertible then ${\displaystyle xy}$ is also invertible with inverse ${\displaystyle (xy)^{-1}=y^{-1}x^{-1}}$. 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.

Kleinfeld's theorem states that any simple non-associative alternative ring is a generalized octonion algebra over its center. ^[6] The structure theory of alternative rings is presented in the book Rings That Are Nearly Associative by Zhevlakov, Slin'ko, Shestakov, and Shirshov. ^[7]

Occurrence

The projective plane over any alternative division ring is a Moufang plane.

Every composition algebra is an alternative algebra, as shown by Guy Roos in 2008: ^[8] A composition algebra A over a field K has a norm n that is a multiplicative homomorphism: ${\displaystyle n(a\times b)=n(a)\times n(b)}$ connecting (A, ×) and (K, ×).

Define the form ( _ : _ ): A × A → K by ${\displaystyle (a:b)=n(a+b)-n(a)-n(b).}$ Then the trace of a is given by (a:1) and the conjugate by a* = (a:1)e – a where e is the basis element for 1. A series of exercises prove that a composition algebra is always an alternative algebra. ^[9]

See also

• Algebra over a field
• Maltsev algebra
• Zorn ring

References

1. Schafer (1995) p. 27
2. Schafer (1995) p. 28
3. Conway, John Horton; Smith, Derek A. (2003). On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A. K. Peters. ISBN   1-56881-134-9. Zbl   1098.17001.
4. Schafer (1995) p. 29
5. Schafer (1995) p. 30
6. Zhevlakov, Slin'ko, Shestakov, Shirshov. (1982) p. 151
7. Zhevlakov, Slin'ko, Shestakov, Shirshov (1982)
8. 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
9. Associative Composition Algebra/Transcendental paradigm#Categorical treatment at Wikibooks

• Schafer, Richard D. (1995). An Introduction to Nonassociative Algebras . New York: Dover Publications. ISBN   0-486-68813-5. Zbl   0145.25601.
• Zhevlakov, K.A.; Slin'ko, A.M.; Shestakov, I.P.; Shirshov, A.I. (1982) [1978]. Rings That Are Nearly Associative. Academic Press. ISBN   0-12-779850-1. MR   0518614. Zbl   0487.17001.

External links

• Zhevlakov, K.A. (2001) [1994], "Alternative rings and algebras", Encyclopedia of Mathematics , EMS Press