Non-associative Algebra

A non-associative algebra (or distributive algebra) over a field (or a commutative ring) K is a K-vector space (or more generally a module) A equipped with a K-bilinear map A × A → A which establishes a binary multiplication operation on A. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd), (a(bc))d and a(b(cd)) may all yield different answers.

While this use of non-associative means that associativity is not assumed, it does not mean that associativity is disallowed. In other words, "non-associative" means "not necessarily associative", just as "noncommutative" means "not necessarily commutative" for noncommutative rings.

Multiplication by elements of A on the left or on the right give rise to left and right K-linear transformations of A given by and . The enveloping algebra of a non-associative algebra A is the subalgebra of the full algebra of K-endomorphisms of A which is generated by the left and right multiplication maps of A. This enveloping algebra is necessarily associative, even though A may be non-associative. In a sense this makes the enveloping algebra "the smallest associative algebra containing A".

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