Non-associative algebra

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[3] (right powers) or xⁿ⁺¹ ≝ xxⁿ[4][5] (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:[6] xy = −yx.
• Jacobi identity:[6][7] (xy)z + (yz)x + (zx)y = 0 or x(yz) + y(zx) + z(xy) = 0 depending on authors.
• Jordan identity:[8][9] (x²y)x = x²(yx) or (xy)x² = x(yx²) depending on authors.
• Alternative:[10][11][12] (xx)y = x(xy) (left alternative) and (yx)x = y(xx) (right alternative).
• Flexible:[13][14] (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:[4][5][15][16][3] 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.

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.[17][lower-alpha 1]
• Commutative implies flexible.
• Anticommutative implies flexible.
• Alternative implies power associative.[lower-alpha 1]
• 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) ≤ 2:

• Jordan identity and commutative together imply power associative.[18][19][20]

If char(K) ≠ 2:

• Right alternative implies power associative.[21][22][23][24]
  • Similarly, left alternative implies power associative.
• Unital and Jordan identity together imply flexible.[25]
• Jordan identity and flexible together imply power associative.[26]
• 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.[27]

If char(K) = 0:

• Third power associative and x⁴ = x²x² (one of the two identities defining fourth power associative) together imply power associative.[28]

If char(K) = 2:

• Commutative and anticommutative are equivalent.

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:[29] [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:[30] 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.

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

${\displaystyle C(A)=\{n\in A\ |\ nr=rn\,\forall r\in A\,\}}$

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.

A derivation on A is a map D with the property

${\displaystyle D(x\cdot y)=D(x)\cdot y+x\cdot D(y)\ .}$

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

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

${\displaystyle L(a):x\mapsto ax;\ \ R(a):x\mapsto xa\ .}$

The associative enveloping algebra or multiplication algebra of A is the associative algebra generated by the left and right linear maps.[29][35] The centroid of A is the centraliser of the enveloping algebra in the endomorphism algebra End_K(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]

• 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:[37]

${\displaystyle Q(a):x\mapsto 2a\cdot (a\cdot x)-(a\cdot a)\cdot x\ }$

or equivalently

${\displaystyle Q(a)=2L^{2}(a)-L(a^{2})\ .}$

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.