Non-associative algebra

A non-associative algebra[1] (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × AA which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. 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.

An algebra is unital or unitary if it has an identity element e with ex = x = xe for all x in the algebra. For example, the octonions are unital, but Lie algebras never are.

The nonassociative algebra structure of A may be studied by associating it with other associative algebras which are subalgebras of the full algebra of K-endomorphisms of A as a K-vector space. Two such are the derivation algebra and the (associative) enveloping algebra, the latter being in a sense "the smallest associative algebra containing A".

Ring-like structures with two binary operations and no other restrictions are a broad class, one which is too general to study. For this reason, the best-known kinds of non-associative algebras satisfy identities, or properties, which simplify multiplication somewhat. These include the following ones.

Let x, y and z denote arbitrary elements of the algebra A over the field K. Let powers to positive (non-zero) integer be recursively defined by x1x and either xn+1xnx[3] (right powers) or xn+1xxn[4][5] (left powers) depending on authors.

The nucleus is the set of elements that associate with all others:[30] that is, the n in A such that

The center of A is the set of elements that commute and associate with everything in A, that is the intersection of

There are several properties that may be familiar from ring theory, or from associative algebras, which are not always true for non-associative algebras. Unlike the associative case, elements with a (two-sided) multiplicative inverse might also be a zero divisor. For example, all non-zero elements of the sedenions have a two-sided inverse, but some of them are also zero divisors.

The free non-associative algebra on a set X over a field K is defined as the algebra with basis consisting of all non-associative monomials, finite formal products of elements of X retaining parentheses. The product of monomials u, v is just (u)(v). The algebra is unital if one takes the empty product as a monomial.[31]

Kurosh proved that every subalgebra of a free non-associative algebra is free.[32]

An algebra A over a field K is in particular a K-vector space and so one can consider the associative algebra EndK(A) of K-linear vector space endomorphism of A. We can associate to the algebra structure on A two subalgebras of EndK(A), the derivation algebra and the (associative) enveloping algebra.

The derivations on A form a subspace DerK(A) in EndK(A). The commutator of two derivations is again a derivation, so that the Lie bracket gives DerK(A) a structure of Lie algebra.[33]

There are linear maps L and R attached to each element a of an algebra A:[34]

The associative enveloping algebra or multiplication algebra of A is the associative algebra generated by the left and right linear maps.[29][35] The centroid of A is the centraliser of the enveloping algebra in the endomorphism algebra EndK(A). An algebra is central if its centroid consists of the K-scalar multiples of the identity.[16]

Some of the possible identities satisfied by non-associative algebras may be conveniently expressed in terms of the linear maps:[36]

The article on universal enveloping algebras describes the canonical construction of enveloping algebras, as well as the PBW-type theorems for them. For Lie algebras, such enveloping algebras have a universal property, which does not hold, in general, for non-associative algebras. The best-known example is, perhaps the Albert algebra, an exceptional Jordan algebra that is not enveloped by the canonical construction of the enveloping algebra for Jordan algebras.