Non-associative algebra explained

A non-associative algebra (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 × A → A 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".

More generally, some authors consider the concept of a non-associative algebra over a commutative ring R: An R-module 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

Z -algebra, so some authors refer to non-associative 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 identities, or properties, which simplify multiplication somewhat.These include the following ones.

Usual properties

Let, and denote arbitrary elements of the algebra over the field .Let powers to positive (non-zero) integer be recursively defined by and either (right powers) or (left powers) depending on authors.

• Unital: there exist an element so that ; in that case we can define .
• Associative: .
• Commutative: .
• Anticommutative: .
• Jacobi identity: or depending on authors.
• Jordan identity: or depending on authors.
• Alternative: (left alternative) and (right alternative).
• Flexible: .
• th power associative with : for all integers so that .
  • Third power associative: .
  • Fourth power associative: (compare with fourth power commutative below).
• Power associative: the subalgebra generated by any element is associative, i.e., th power associative for all .
• th power commutative with : for all integers so that .
  • Third power commutative: .
  • Fourth power commutative: (compare with fourth power associative above).
• Power commutative: the subalgebra generated by any element is commutative, i.e., th power commutative for all .
• Nilpotent of index : the product of any elements, in any association, vanishes, but not for some elements: and there exist elements so that for a specific association.
• Nil of index : power associative and and there exist an element so that .

Relations between properties

For of any characteristic:

• Associative implies alternative.
• Any two out of the three properties left alternative, right alternative, and flexible, imply the third one.
  • Thus, alternative implies flexible.
• Alternative implies Jordan identity.
• Commutative implies flexible.
• Anticommutative implies flexible.
• Alternative implies power associative.
• Flexible implies third power associative.
• Second power associative and second power commutative are always true.
• Third power associative and third power commutative are equivalent.
• th power associative implies th power commutative.
• Nil of index 2 implies anticommutative.
• Nil of index 2 implies Jordan identity.
• Nilpotent of index 3 implies Jacobi identity.
• Nilpotent of index implies nil of index with .
• Unital and nil of index are incompatible.

If or :

• Jordan identity and commutative together imply power associative.

If :

• Right alternative implies power associative.
  • Similarly, left alternative implies power associative.
• Unital and Jordan identity together imply flexible.
• Jordan identity and flexible together imply power associative.
• Commutative and anticommutative together imply nilpotent of index 2.
• Anticommutative implies nil of index 2.
• Unital and anticommutative are incompatible.

If :

• Unital and Jacobi identity are incompatible.

If :

• Commutative and (one of the two identities defining fourth power associative) together imply power associative.

If :

• Third power associative and (one of the two identities defining fourth power associative) together imply power associative.

If :

• Commutative and anticommutative are equivalent.

Associator

See main article: Associator.

[ ⋅ , ⋅ , ⋅ ] : A x A x A \to A given by

.

It measures the degree of nonassociativity of

A , and can be used to conveniently express some possible identities satisfied by A.

Let, and denote arbitrary elements of the algebra.

• Associative: .
• Alternative: (left alternative) and (right alternative).
  • It implies that permuting any two terms changes the sign: ; the converse holds only if .
• Flexible: .
  • It implies that permuting the extremal terms changes the sign: ; the converse holds only if .
• Jordan identity: or depending on authors.
• Third power associative: .

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