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power-associative algebra

Power-associative algebras

Power-associative algebras

Idea

In an associative algebra, or more generally in a semigroup, there is no difficulty in defining x nx^n for xx any element and nn any positive natural number (as well as x 0x^0 when the algebra has an identity element). In a nonassociative algebra, or more generally a magma, we can have many definitions, starting with x 3x^3 (which could be (xx)x(x x) x but could be x(xx)x (x x) instead).

However, it may be that all possible ways of parenthesizing the expression for x nx^n are equal. In this case, we call the algebra power-associative.

Definition

Let 𝒞\mathcal{C} be a multicategory, and let AA be a magma object in 𝒞\mathcal{C}, that is an object AA equipped with a bimorphism (binary multimorphism) m:A,AAm\colon A, A \to A. Consider the submulticategory of 𝒞\mathcal{C} generated by AA and mm; for each natural number nn, it has a family of nn-ary multimorphisms whose size is the Catalan number C(n)C(n) of nn.

The magma object AA is power-associative if, for each nn, all of these nn-ary multimorphisms are equal. Traditionally, we make no requirement for n=0n = 0, since C(0)=0C(0) = 0, but arguably we should require a unique 00-ary multimorphism in the subalgebra too; then we get a power-associative magma with an identity element.

If 𝒞\mathcal{C} is Set, then a magma object is simply a magma, and we have a power-associative magma. If 𝒞\mathcal{C} is KK-Mod for KK a commutative ring, then a magma object is a nonassociative algebra over KK, and we have a power-associative algebra.

Properties

The submagma of every power-associative invertible magma MM generated by an element aMa \in M is a cyclic group. This means in particular there is a \mathbb{Z}-action on MM () ():M×M(-)^{(-)}:M\times\mathbb{Z}\to M called the power.

Examples

Every associative algebra or semigroup is of course power-associative. More generally, every alternative algebra is also power-associative. Every Jordan algebra, although not necessarily alternative, is power-associative. Every Cayley–Dickson algebra, even beyond the octonions (the last alternative one), is power-associative.

Last revised on May 25, 2021 at 18:53:45. See the history of this page for a list of all contributions to it.