Non-associative algebra

A non-associative algebra (or distributive algebra) is an where the  is not assumed to be. That is, an A is a non-associative algebra over a  K if it is a  over K and is equipped with a K- binary multiplication operation A × A → A which may or may not be associative. Examples include s, s, the s, and three-dimensional Euclidean space equipped with the 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 s.

An algebra is  or unitary if it has an I with Ix = x = xI for all x in the algebra. For example, the s are unital, but s never are.

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

More generally, some authors consider the concept of a non-associative algebra over a R: An  equipped with an R-bilinear binary multiplication operation. If a structure obeys all of the ring axioms apart from associativity (for example, any R-algebra), then it is naturally a $$\mathbb{Z}$$-algebra, so some authors refer to non-associative $$\mathbb{Z}$$-algebras as non-associative rings.

Algebras satisfying identities
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 which simplify multiplication somewhat. These include the following identities.

In the list, x, y and z denote arbitrary elements of an algebra.
 * : (xy)z = x(yz).
 * : xy = yx.
 * : xy = −yx.
 * : (xy)z + (yz)x + (zx)y = 0.
 * : (xy)(x2) = x(y(x2)).
 * : For all x, any three nonnegative powers of x associate. That is if a, b and c are nonnegative powers of x, then a(bc) = (ab)c. This is equivalent to saying that xm xn = xn+m for all non-negative integers m and n.
 * : (xx)y = x(xy) and (yx)x = y(xx).
 * : x(yx) = (xy)x.
 * Elastic: Flexible and (xy)(xx) = x(y(xx)), x(xx)y = (xx)(xy).

These properties are related by
 * 1) associative implies alternative implies power associative;
 * 2) associative implies Jordan identity implies power associative;
 * 3) Each of the properties associative, commutative, anticommutative, Jordan identity, and Jacobi identity individually imply flexible, which implies power associative.
 * 4) For a field with characteristic not two, being both commutative and anticommutative implies the algebra is just {0}.
 * 5) alternative implies flexible. If the algebra is not commutative, any two out of the three following properties: left-alternative, right-alternative, flexible, imply the third one.

Associator
The associator on A is the K- $$[\cdot,\cdot,\cdot] : A \times A \times A \to A$$ given by


 * $$[x,y,z] = (xy)z - x(yz).\,$$

It measures the degree of nonassociativity of $$A$$, and can be used to conveniently express some possible identities satisfied by A.


 * Associative: the associator is identically zero;
 * Alternative: the associator is, interchange of any two terms changes the sign;
 * Flexible: $$[x,y,x] = 0$$;
 * Jordan: $$[x, y, x^2] = 0$$.

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


 * $$ [n,A,A] = [A,n,A] = [A,A,n] = \{0\} \ . $$

The nucleus is an associative subring of A.

Center
The center of a A is the set of elements that commute and associate with everything in A, that is the intersection of
 * $$ C(A) = \{ n \in A \ | \ nr=rn \, \forall r \in A \, \} $$

with the nucleus. It turns out that for elements of C(A) it is enough that two of the sets $$([n,A,A], [A,n,A], [A,A,n])$$ are $$\{0\}$$ for the third to also be the zero set.

Examples

 * R3 with multiplication given by the is an example of an algebra which is anticommutative and not associative. The cross product also satisfies the Jacobi identity.
 * s are algebras satisfying anticommutativity and the Jacobi identity.
 * Algebras of s on a (if K is R or the s C) or an  (for general K);
 * s are algebras which satisfy the commutative law and the Jordan identity.
 * Every associative algebra gives rise to a Lie algebra by using the as Lie bracket. In fact every Lie algebra can either be constructed this way, or is a subalgebra of a Lie algebra so constructed.
 * Every associative algebra over a field of 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.
 * s are algebras satisfying the alternative property. The most important examples of alternative algebras are the (an algebra over the reals), and generalizations of the octonions over other fields. 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.
 * s, are those algebras satisfying the power-associative identity. Examples include all associative algebras, all alternative algebras, Jordan algebras, and the s.
 * The algebra over R, which was an experimental algebra before the adoption of  for.

More classes of algebras:


 * s. These include most of the algebras of interest to, such as the , , and over a given . Graded algebras can be generalized to s.
 * s, in which multiplicative inverses exist. The finite-dimensional alternative division algebras over the field of real numbers have been classified. They are the s (dimension 1), the s (dimension 2), the s (dimension 4), and the s (dimension 8). The quaternions and octonions are not commutative. Of these algebras, all are associative except for the octonions.
 * s, 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 s (where K is R), which begin with:
 * C (a commutative and associative algebra);
 * the s H (an associative algebra);
 * the s (an );
 * the s, and the infinite sequence of Cayley-Dickson algebras (s).
 * The s are considered in . They carry two multiplications, turning them into commutative algebras and Lie algebras in different ways.
 * s are non-associative algebras used in mathematical genetics.
 * s

Properties
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. For example, all non-zero elements of the s have a two-sided inverse, but some of them are also zero divisors.

Free non-associative algebra
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.

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

Associated algebras
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.

Derivation algebra
A  on A is a map D with the property


 * $$D(x \cdot y) = D(x) \cdot y + x \cdot D(y) \ . $$

The derivations on A form a subspace DerK(A) in EndK(A). The of two derivations is again a derivation, so that the  gives DerK(A) a structure of.

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


 * $$L(a) : x \mapsto ax ; \ \ R(a) : x \mapsto xa \ . $$

The associative enveloping algebra or multiplication algebra of A is the associative algebra generated by the left and right linear maps. 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.

Some of the possible identities satisfied by non-associative algebras may be conveniently expressed in terms of the linear maps:
 * Commutative: each L(a) is equal to the corresponding R(a);
 * Associative: any L commutes with any R;
 * Flexible: every L(a) commutes with the corresponding R(a);
 * Jordan: every L(a) commutes with R(a2);
 * Alternative: every L(a)2 = L(a2) and similarly for the right.

The quadratic representation Q is defined by


 * $$Q(a) : x \mapsto 2a \cdot (a \cdot x) - (a \cdot a) \cdot x \ $$

or equivalently


 * $$Q(a) = 2 L^2(a) - L(a^2) \ . $$

The article on s 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, an exceptional that is not enveloped by the canonical construction of the enveloping algebra for Jordan algebras.