Alternative algebra - Knowlpedia

Definition

An algebra $A$ is alternative if it satisfies the left and right alternative laws

x(xy)=(xx)y, \qquad (yx)x=y(xx)

for all $x,y\in A$ . In terms of the associator, these are $[x,x,y]=0$ and $[y,x,x]=0$ .

What alternativity provides

Artin's theorem says that the subalgebra generated by any two elements of an alternative algebra is associative. Consequently, expressions involving only two fixed elements may be reassociated freely. In a unital alternative algebra this makes powers $x^n$ unambiguous and permits the familiar calculations inside the subalgebra generated by $x$ and $y$ .

The Moufang identities, such as

(xy)(zx)=x(yz)x,

also hold in every alternative algebra. They replace many associative rearrangements when three elements are involved.

Associator symmetry

Over a field of characteristic different from $2$ , linearizing the alternative laws shows that the associator is alternating: it changes sign when two arguments are exchanged and vanishes when two arguments agree. This description needs care in small characteristic, while the two displayed alternative laws make sense over any base ring.

Examples and nonexamples

Every associative algebra is alternative.
Every unital composition algebra is alternative.
The octonions are the standard alternative, nonassociative division algebra.
A general Jordan algebra is power-associative but need not be alternative.

References

Richard D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, 1966. Project Gutenberg edition. Relevant: Chapter III, §§1–2.
John C. Baez, “The Octonions,” Bulletin of the American Mathematical Society 39 (2002), 145–205. DOI record. Relevant: §2.1.