Non-associative algebra

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

Usual properties

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 x¹ ≝ x and either xⁿ⁺¹ ≝ xⁿx (right powers) or xⁿ⁺¹ ≝ xxⁿ (left powers) depending on authors.

• Unital: there exist an element e so that ex = x = xe; in that case we can define x⁰ ≝ e.
• Associative: (xy)z = x(yz).
• Commutative: xy = yx.
• Anticommutative: xy = −yx.
• Jacobi identity: (xy)z + (yz)x + (zx)y = 0 or x(yz) + y(zx) + z(xy) = 0 depending on authors, but these are equivalent for anticommutative algebras, and Lie algebras are anticommutative algebras obeying the Jacobi identity.
• Jordan identity: (x²y)x = x²(yx) or (xy)x² = x(yx²) depending on authors, but these are equivalent for commutative algebras, and Jordan algebras are commutative algebras obeying the Jordan identity.
• Alternative: (xx)y = x(xy) (left alternative) and (yx)x = y(xx) (right alternative).
• Flexible: (xy)x = x(yx).
• nth power associative with n ≥ 2: x^n−kx^k = xⁿ for all integers k so that 0 < k < n.
  • Third power associative: x²x = xx².
  • Fourth power associative: x³x = x²x² = xx³ (compare with fourth power commutative below).
• Power associative: the subalgebra generated by any element is associative, i.e., nth power associative for all n ≥ 2.
• nth power commutative with n ≥ 2: x^n−kx^k = x^kx^n−k for all integers k so that 0 < k < n.
  • Third power commutative: x²x = xx².
  • Fourth power commutative: x³x = xx³ (compare with fourth power associative above).
• Power commutative: the subalgebra generated by any element is commutative, i.e., nth power commutative for all n ≥ 2.
• Nilpotent of index n ≥ 2: the product of any n elements, in any association, vanishes, but not for some n−1 elements: x₁x₂…x_n = 0 and there exist n−1 elements so that y₁y₂…y_n−1 ≠ 0 for a specific association.
• Nil of index n ≥ 2: power associative and xⁿ = 0 and there exist an element y so that yⁿ⁻¹ ≠ 0.

Relations between properties

For K 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.
• nth power associative implies nth 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 n implies nil of index N with 2 ≤ N ≤ n.
• Unital and nil of index n are incompatible.

If K ≠ GF(2) or dim(A) ≤ 3:

• Jordan identity and commutative together imply power associative.

If char(K) ≠ 2:

• 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 char(K) ≠ 3:

• Unital and Jacobi identity are incompatible.

If char(K) ∉ {2,3,5}:

• Commutative and x⁴ = x²x² (one of the two identities defining fourth power associative) together imply power associative.

If char(K) = 0:

• Third power associative and x⁴ = x²x² (one of the two identities defining fourth power associative) together imply power associative.

If char(K) = 2:

• Commutative and anticommutative are equivalent.

Associator

The associator on A is the K-multilinear map ${\displaystyle [\cdot ,\cdot ,\cdot ]:A\times A\times A\to A}$ given by

[x,y,z] = (xy)z − x(yz).

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

Let x, y and z denote arbitrary elements of the algebra.

• Associative: [x,y,z] = 0.
• Alternative: [x,x,y] = 0 (left alternative) and [y,x,x] = 0 (right alternative).
  • It implies that permuting any two terms changes the sign: [x,y,z] = −[x,z,y] = −[z,y,x] = −[y,x,z]; the converse holds only if char(K) ≠ 2.
• Flexible: [x,y,x] = 0.
  • It implies that permuting the extremal terms changes the sign: [x,y,z] = −[z,y,x]; the converse holds only if char(K) ≠ 2.
• Jordan identity: [x²,y,x] = 0 or [x,y,x²] = 0 depending on authors.
• Third power associative: [x,x,x] = 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 is the set of elements that commute and associate with everything in A, that is the intersection of $${\displaystyle C(A)=\{n\in A|nr=rn\mathrm{\forall }r\in A\}}$$ with the nucleus. It turns out that for elements of C(A) it is enough that two of the sets ${\displaystyle ([n,A,A],[A,n,A],[A,A,n])}$ are ${\displaystyle \{0\}}$ for the third to also be the zero set.