Power-associative algebra

Definition

A nonassociative algebra $A$ is power-associative if the subalgebra generated by each single element $x\in A$ is associative. Equivalently, every product made only from copies of $x$ is independent of parenthesization, so the powers $x^n$ are unambiguous and satisfy

x^m x^n=x^{m+n} \qquad(m,n\geq1).

Scope of the condition

Power associativity controls expressions in one element only. It does not assert $(xy)z=x(yz)$ for unrelated $x,y,z$ , and therefore does not make $A$ an associative algebra. It is precisely strong enough to make polynomial expressions $p(x)$ , nilpotence of an element, and idempotents meaningful in the familiar way.

Jordan and alternative algebras

Every Jordan algebra in the usual characteristic-different-from- $2$ setting is power-associative: the Jordan identity forces the powers generated by one element to associate. Every alternative algebra is also power-associative, since the subalgebra generated by any two elements is associative, a stronger conclusion.

The qualifier cannot simply be omitted when working with an arbitrary nonassociative algebra. Without it, even $x^3$ may depend on whether one means $(xx)x$ or $x(xx)$ .

Characteristic caveat

Criteria obtained by polarizing identities can require restrictions on the characteristic of the base field. The intrinsic definition by associative one-generated subalgebras does not depend on a chosen list of polarized identities and is therefore the safest general formulation.

References

Richard D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, 1966, Chapters II, IV, and V. Publisher record.
Kevin McCrimmon, A Taste of Jordan Algebras, Springer, 2004, Chapter 1. Publisher record.