Non-associative algebra

Non-associative algebra
This article is about a particular non-associative structure known as a non-associative algebra. See also the article about non-associativity in general.

A non-associative algebra[1] (or distributive algebra) over a field (or a ring) K is a K-vector space (or more generally a module[2]) A equipped with a K-bilinear map A × A → A. There are left and right multiplication maps L_a : x \mapsto ax and R_a : x \mapsto xa. The enveloping algebra of A is the subalgebra of all K-endomorphisms of A generated by the multiplication maps.

An algebra is unital or unitary if it has a unit or identity element I with Ix = x = xI for all x in the algebra.

Examples

The best-known kinds of non-associative algebras are those that are nearly associative—that is, in which some simple equation constrains the differences between different ways of associating multiplication of elements. These include:

  • Jordan algebras which are commutative and satisfy the Jordan property (xy)x2 = x(yx2) and also xy = yx
    • every associative algebra over a field of characteristic other than 2 gives rise to a Jordan algebra by defining a new multiplication x*y = (1/2)(xy + yx). In contrast to the Lie algebra case, not every Jordan algebra can be constructed this way. Those that can are called special.
  • Alternative algebras, which require that (xx)y = x(xy) and (yx)x = y(xx). The most important examples are the octonions (an algebra over the reals), and generalizations of the octonions over other fields. (Obviously all associative algebras are alternative.) Up to isomorphism the only finite-dimensional real alternative, division algebras (see below) are the reals, complexes, quaternions and octonions.
  • Power-associative algebras, which require that xmxn = xm+n, where m ≥ 1 and n ≥ 1. (Here we formally define xn recursively as x(xn−1).) Examples include all associative algebras, all alternative algebras, and the sedenions.

These properties are related by 1) associative implies alternative implies power associative; 2) commutative and associative implies Jordan implies power associative. None of the converse implications hold.

More classes of algebras:

  • Division algebras, in which multiplicative inverses exist or division can be carried out. The finite-dimensional alternative division algebras over the field of real numbers can be classified nicely. They are the real numbers (dimension 1), the complex numbers (dimension 2), the quaternions (dimension 4), and the octonions (dimension 8).
  • Quadratic algebras, which require that xx = re + sx, for some elements r and s in the ground field, and e a unit for the algebra. Examples include all finite-dimensional alternative algebras, and the algebra of real 2-by-2 matrices. Up to isomorphism the only alternative, quadratic real algebras without divisors of zero are the reals, complexes, quaternions, and octonions.
  • The Cayley–Dickson algebras (where K is R), which begin with:

References

  1. ^ Richard D. Schafer, An Introduction to Nonassociative Algebras (1996) ISBN 0-486-68813-5 Gutenberg eText
  2. ^ See page 1, from Richard S. Pierce. Associative algebras. Springer. Graduate texts in mathematics, 88.

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