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power-associative algebra

Context

Algebra

Algebraic theories

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Algebras and modules

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Higher algebras

  • symmetric monoidal (โˆž,1)-category of spectra

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Model category presentations

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Geometry on formal duals of algebras

Theorems

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 ๐’ž\mathcal{C} be a multicategory, and let AA be a magma object in ๐’ž\mathcal{C}, that is an object AA equipped with a bimorphism (binary multimorphism) m:A,Aโ†’Am\colon A, A \to A. Consider the submulticategory of ๐’ž\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 ๐’ž\mathcal{C} is Set, then a magma object is simply a magma, and we have a power-associative magma. If ๐’ž\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.

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 Cayleyโ€“Dickson algebra, even beyond the octonions (the last alternative one), is power-associative.

Last revised on September 5, 2016 at 06:09:25. See the history of this page for a list of all contributions to it.