Non-Associative Algebras and the Property Ladder

Introduction

This article stands in the category Linear Algebras, immediately before Division Algebras, and it studies the identities that lie between associativity and its failure. The corpus default is the commutative ring with identity; the objects here are algebras over a field, in the sense of Algebras, above, and the base field is assumed of characteristic not 2 or 3 wherever the symmetrised or the linearised form of an identity is used. The guiding question is not what a non-associative algebra is — that is a definition — but which identities survive when associativity is dropped, and in which order.

Everything is stated as a polynomial identity in the elements of the algebra. The article therefore needs no norm, no inner product and no topology: the octonions are introduced here by structure constants, so that the identities can be checked directly on the multiplication table. The two results that the neighbouring articles own are quoted, not repeated: the Hurwitz theorem and the classification of the normed division algebras belong to Normed Division Algebras and the Hurwitz Theorem, in Part II, where the norm and the form are available, and the Jordan identity belongs to Jordan Algebras, later in this Part; each is cited where the ladder reaches it.

Algebras, the Associator and Linearisation

Definition. Let $k$ be a field. An algebra over $k$ is a $k$-module $A$ with a bilinear product $A \times A \to A$. The algebra is associative if $(xy)z = x(yz)$ for all $x, y, z \in A$, commutative if $xy = yx$ for all $x, y$, and unital if it has an identity, which is then written $1 \neq 0$; a non-associative algebra is an algebra in which the product need not be associative, and the term does not exclude the associative case.

Definition. The associator of $x, y, z \in A$ is

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

The associator is a trilinear function of its three arguments, and the algebra is associative exactly when the associator vanishes identically. Every identity considered in this article is an identity between products of three or four elements, and is therefore an identity in the associator together with the two products.

Lemma (linearisation). Let $A$ be an algebra over a field of characteristic not 2 and let $[\cdot,\cdot,\cdot]$ be its associator. Then $A$ is left alternative, that is $x^2y = x(xy)$ for all $x, y$, if and only if $[x,z,y] = -[z,x,y]$ for all $x, y, z$; and $A$ is right alternative, that is $yx^2 = (yx)x$, if and only if $[y,z,x] = -[y,x,z]$ for all $x, y, z$.

Proof. The left alternative law says $[x,x,y] = 0$ for all $x, y$, and the right alternative law says $[y,x,x] = 0$. Replace $x$ by $x + z$ in the first: by trilinearity,

$$ 0 = [x+z, x+z, y] = [x,x,y] + [x,z,y] + [z,x,y] + [z,z,y] = [x,z,y] + [z,x,y] , $$

which gives the stated antisymmetry in the first two arguments; the second statement is the same computation in the last two arguments. Conversely the antisymmetry with $z = x$ gives $2[x,x,y] = 0$, hence $[x,x,y] = 0$ in characteristic not 2.

Corollary. An algebra over a field of characteristic not 2 is alternative, that is both left and right alternative, if and only if its associator is an alternating function of its arguments: $[x,y,z]$ changes sign under every transposition of two arguments.

Proof. An alternating trilinear function vanishes whenever two arguments coincide, which is alternativity in the two forms of the lemma; conversely the antisymmetries of the lemma are the transpositions of the first two and of the last two arguments, and the remaining transposition is the composite of them.

The Ladder of Identities

Definition. Let $A$ be an algebra over a field. The identities below are listed from the strongest to the weakest; the precise implications between them are the theorem that follows, and two of the entries, flexibility and power-associativity, are not comparable with one another, as the examples of §What the Ladder Does Not Decide show in both directions.

$A$ is associative if $[x,y,z] = 0$ for all $x, y, z$.
$A$ is left alternative if $[x,x,y] = 0$ and right alternative if $[y,x,x] = 0$; $A$ is alternative if it is both, that is if $x(xy) = x^2y$ and $(yx)x = yx^2$ for all $x, y$.
$A$ is flexible if $[x,y,x] = 0$ for all $x, y$, that is $x(yx) = (xy)x$.
$A$ is power-associative if the subalgebra generated by any one element is associative, equivalently if $x^px^q = x^{p+q}$ for all $p, q \geq 1$ and all $x$, the powers being defined by $x^1 = x$ and $x^{n+1} = x^nx$.
$A$ satisfies the Moufang identities if for all $x, y, z$

$$ (xyx)z = x(y(xz)), \qquad z(xyx) = ((zx)y)x, \qquad (xy)(zx) = x(yz)x . $$

$A$ satisfies third-power associativity if $x^2x = xx^2$ for all $x$, the weakest identity of the list: it is flexibility with $y = x$, so every algebra that is flexible, and hence every algebra above it on the ladder, satisfies it.

Theorem (the order of the ladder). Let $A$ be an algebra over a field of characteristic not 2 or 3. Then

$$ \text{associative} \implies \text{alternative} \implies \text{flexible} , \qquad \text{alternative} \implies \text{power-associative} , \qquad \text{alternative} \implies \text{Moufang identities} . $$

Proof. Associativity gives alternativity by definition, and alternativity gives flexibility because the associator is then alternating and has its first and third arguments equal in $[x,y,x]$, so that $[x,y,x] = -[x,y,x]$ and the associator vanishes in characteristic not 2. That an alternative algebra is power-associative is Artin's theorem: in an alternative algebra the subalgebra generated by any two elements is associative, cited from the literature; the subalgebra generated by one element is then associative, which is power-associativity. That an alternative algebra satisfies the three Moufang identities is likewise standard and is cited; the middle one is the case in which the three arguments are paired in the two possible ways, and the proof uses Artin's theorem on two generators.

Remark. The ladder is a chain of implications and not a chain of equivalences, and the implications are strict at every step that is separated below: the octonions are alternative and not associative, the algebra $Q$ of §What the Ladder Does Not Decide is power-associative and not alternative, and the algebra $P$ there is flexible and not power-associative; the algebra $S$ there is power-associative and not flexible. The order matters for the corpus because an alternative algebra is the weakest classical algebra in which a calculus of powers — and hence a notion of $x^n$ for a single element, which is what a norm later uses — is still available, while associativity is unavailable.

The Octonions

Definition. Let $k$ be a field of characteristic not 2. The octonion algebra $\mathbb{O}$ is the $k$-algebra with basis $1, e_1, \ldots, e_7$ and product with unit $1$, with

$$ e_i^2 = -1 \quad (1 \leq i \leq 7), \qquad e_ie_j = -e_je_i \quad (i \neq j), $$

and with the products of the distinct-basis pairs given by the seven oriented triples

triple	$(1,2,3)$	$(1,4,5)$	$(1,7,6)$	$(2,4,6)$	$(2,5,7)$	$(3,4,7)$	$(3,6,5)$
product	$e_1e_2 = e_3$	$e_1e_4 = e_5$	$e_1e_7 = e_6$	$e_2e_4 = e_6$	$e_2e_5 = e_7$	$e_3e_4 = e_7$	$e_3e_6 = e_5$

in the sense that a triple $(i,j,k)$ in the table means

$$ e_ie_j = e_k, \qquad e_je_k = e_i, \qquad e_ke_i = e_j , $$

the products in the reverse orderings being their negatives, $e_je_i = -e_k$ and so on. With this convention the table defines the product of every pair of basis elements, the orientation of each triple being part of the definition; each pair of distinct basis elements lies in exactly one triple, so the rules do not conflict. The algebra is eight-dimensional, not commutative and not associative.

Theorem. The octonion algebra $\mathbb{O}$ is alternative, flexible, power-associative and satisfies the three Moufang identities.

Proof. The associator is trilinear, so an identity between associators holds for all elements as soon as it holds on all $8^3$ triples of basis elements, and alternativity is equivalent to the antisymmetry of the associator by the corollary above. The associator vanishes as soon as one of its arguments is the identity $1$, and the remaining cases are the $7 \cdot 6 \cdot 5 = 210$ ordered triples of distinct elements of $e_1, \ldots, e_7$, each evaluated from the table; the computation gives

$$ [e_i,e_j,e_k] = -[e_j,e_i,e_k] = -[e_i,e_k,e_j] \qquad \text{for all } i, j, k , $$

so the associator is alternating and $\mathbb{O}$ is alternative; it is flexible, power-associative and satisfies the Moufang identities by the theorem on the order of the ladder, since an alternative algebra is all three, the first by the antisymmetry just displayed and the last two by Artin's theorem and the Moufang computation cited there.

Theorem. The octonion algebra $\mathbb{O}$ is not associative, and the associator is nonzero.

Proof. From the table $e_1e_2 = e_3$, $e_1e_7 = e_6$ and $e_3e_4 = e_7$, so

$$ (e_1e_2)e_4 = e_3e_4 = e_7 , \qquad e_1(e_2e_4) = e_1e_6 = -e_6e_1 = -e_7 , $$

using $e_2e_4 = e_6$ and the anticommutativity of $e_1$ and $e_6$. Hence

$$ [e_1,e_2,e_4] = (e_1e_2)e_4 - e_1(e_2e_4) = e_7 + e_7 = 2e_7 \neq 0 , $$

and the algebra is not associative. It is also not commutative, since $e_2e_1 = -e_3 \neq e_3 = e_1e_2$.

Remark. The seven triples of the definition are the lines of the Fano plane with a chosen orientation, and the orientation is the part of the definition that matters: an arbitrary orientation of the seven lines gives an algebra that need not be alternative. The construction of $\mathbb{O}$ by doubling, and the description of the doubling as the Cayley–Dickson construction, belong to Normed Division Algebras and the Hurwitz Theorem, in Part II, where the norm is available; the definition by structure constants above is the same algebra and is the one that needs no form. That $\mathbb{O}$ is a division algebra, and that the four normed division algebras are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ with no others, is the Hurwitz theorem and is cited, not proved here; it belongs to that Part II article, with the classification over $\mathbb{R}$ of Division Algebras, later in this category.

The Ladder as the Doubling Proceeds

The four classical algebras over $\mathbb{R}$ are built one from the next, and the ladder records what is lost at each step and what is kept.

algebra	dimension	commutative	associative	alternative	flexible	power-associative	Moufang
$\mathbb{R}$	1	yes	yes	yes	yes	yes	yes
$\mathbb{C}$	2	yes	yes	yes	yes	yes	yes
$\mathbb{H}$	4	no	yes	yes	yes	yes	yes
$\mathbb{O}$	8	no	no	yes	yes	yes	yes

Theorem. The table is correct: the properties listed hold in each algebra, and each of the two failures occurs for the reason shown.

Proof. The reals and the complex numbers are associative and commutative, hence satisfy every weaker identity. The quaternions are associative and not commutative, as in Non-Commutative Domains, above. The octonions are alternative, flexible, power-associative and Moufang by the theorems of the previous section, and neither associative nor commutative. The failure of commutativity appears at the step from $\mathbb{C}$ to $\mathbb{H}$, with $ij = -ji$, and the failure of associativity appears at the step from $\mathbb{H}$ to $\mathbb{O}$, with $[e_1,e_2,e_4] = 2e_7$.

Remark. The step after the octonions, the sixteen-dimensional algebra obtained by one further doubling, is not alternative and has zero divisors; it is outside the ladder of this article and is cited from the literature. What the doubling does to the remaining structure, and where it stops, is the content of the Hurwitz theorem, cited above.

What the Ladder Does Not Decide

The implications of the ladder are one-directional, and the following two algebras show that no converse holds.

Example (flexible, not power-associative). Let $P$ be the commutative algebra with basis $u, v$ and product

$$ u^2 = v, \qquad uv = vu = v, \qquad v^2 = 0 . $$

Then $P$ is commutative, hence flexible. But the two bracketings of the fourth power of $u$ disagree:

$$ (u^2)(u^2) = v^2 = 0 , \qquad ((u^2)u)u = (vu)u = vu = v \neq 0 , $$

so $P$ is not power-associative. Hence flexibility does not imply power-associativity, and the ladder is not a chain of equivalences at that step.

Example (power-associative, not alternative). Let $Q$ be the algebra $M_2(k)$ of two-by-two matrices over a field of characteristic not 2, with the symmetrised product

$$ x \bullet y = \tfrac{1}{2}(xy + yx) , $$

which is commutative and bilinear. The subalgebra generated by one element $z$ is associative: writing $z^{\circ n}$ for the $n$-th power for the product $\bullet$, induction gives $z^{\circ n} = z^n$, the ordinary power of the matrix $z$, because $z^{\circ(n+1)} = z^{\circ n} \bullet z = \tfrac{1}{2}(z^nz + zz^n) = z^{n+1}$; hence $z^{\circ m} \bullet z^{\circ n} = \tfrac{1}{2}(z^mz^n + z^nz^m) = z^{m+n}$, and the subalgebra generated by $z$ is the image of the polynomial algebra $k[t]$ under $t \mapsto z$, which is associative. So $Q$ is power-associative. It is not alternative: with $x = E_{11}$ and $y = E_{12}$,

$$ (x \bullet x) \bullet y = E_{11} \bullet E_{12} = \tfrac{1}{2}E_{12}, \qquad x \bullet (x \bullet y) = E_{11} \bullet \tfrac{1}{2}E_{12} = \tfrac{1}{4}E_{12} , $$

and the two differ because the characteristic is not 2. Hence power-associativity does not imply alternativity.

Example (power-associative, not flexible). Let $S$ be the algebra with basis $e, f$ over a field $k$ and product

$$ e^2 = e, \qquad f^2 = f, \qquad ef = e + f, \qquad fe = 0 . $$

An element $z = \alpha e + \beta f$ satisfies

$$ z^2 = \alpha^2 e + \beta^2 f + \alpha\beta(ef) + \alpha\beta(fe) = (\alpha^2 + \alpha\beta)e + (\beta^2 + \alpha\beta)f = (\alpha + \beta) z , $$

a scalar multiple of $z$; hence $z^n = (\alpha+\beta)^{n-1} z$ for every $n \geq 1$ and for every bracketing, since the product of two scalar multiples of $z$ is again a scalar multiple of $z$ and the scalar $(\alpha+\beta)^{p-1}(\alpha+\beta)^{q-1}(\alpha+\beta) = (\alpha+\beta)^{p+q-1}$ is the one required. So $S$ is power-associative. It is not flexible:

$$ [f,e,f] = (fe)f - f(ef) = 0 - f(e+f) = -f^2 = -f \neq 0 . $$

Hence power-associativity does not imply flexibility, and with the algebra $P$ above the two conditions are incomparable.

Remark. The four examples of the article together separate the steps of the ladder that are separated in this article: $\mathbb{O}$ is alternative and not associative, $P$ is flexible and not alternative, $Q$ is power-associative and not alternative, and $S$ is power-associative and not flexible, so that flexibility and power-associativity are incomparable in both directions. The remaining pair of the middle of the ladder, the Moufang identities and alternativity, is not separated here: an algebra satisfying the Moufang identities without being alternative is not exhibited here, and that pair is left to the literature.

The Neighbouring Theories

Remark. The ladder reaches two theories that this article does not develop. The first is the one of Jordan Algebras, later in this Part: over a field of characteristic not 2 one symmetrises the product of any algebra by

$$ x \bullet y = \tfrac{1}{2}(xy + yx) , $$

which is commutative in every case and, for $\mathbb{H}$ and for $\mathbb{O}$, satisfies the Jordan identity; the identity itself, and the class of Jordan algebras, belong to that article and are not stated here. The second is the one of Division Algebras, later in this category, and of Normed Division Algebras and the Hurwitz Theorem, in Part II: the norm of $\mathbb{O}$, the composition identity it satisfies, and the Hurwitz theorem that no other algebra over $\mathbb{R}$ carries such a norm, are stated there. What this article contributes to both is the ladder: it is exactly the list of identities that remains available at the rung of $\mathbb{O}$, that is, alternativity, flexibility, power-associativity and the Moufang identities, with associativity gone.

Remark. The ladder is also the reason the phrase "non-associative" is not the name of a class of algebras with a structure theory: the class of all non-associative algebras is too large for one, and the corpus never uses the phrase except as the background of the rungs above. An algebra in this category is studied through the subset of the ladder it satisfies.

Summary

An algebra over a field need not be associative, and the identities between associativity and its failure form a ladder: associativity, alternativity (left and right), flexibility, power-associativity and the Moufang identities, related by the implications of the theorem on the order of the ladder, with Artin's theorem supplying power-associativity from alternativity. The octonions, defined by the structure constants of the seven oriented triples of the Fano plane, are alternative, flexible, power-associative and Moufang, and are neither associative nor commutative; the quaternions are associative and not commutative; the reals and the complex numbers satisfy every identity of the ladder. The implications of the ladder are strict at each of the steps separated below: $\mathbb{O}$ separates associativity from alternativity, the algebra $Q$ is power-associative and not alternative and so separates alternativity from power-associativity, the algebra $P$ is flexible and not power-associative, and the algebra $S$ is power-associative and not flexible, which shows that flexibility and power-associativity are incomparable in both directions. The Hurwitz theorem and the classification of the normed division algebras belong to Part II; the Jordan algebras are a neighbour of this article, later in this Part. Both are cited rather than developed.

Summary of Notation

Symbol	Meaning
$k$	A field, of characteristic not 2 or 3 where the linearised identities are used
$A$	An algebra over a field, not assumed associative; the base of this article
$P$, $Q$, $S$	The algebras of the counterexamples of §What the Ladder Does Not Decide: flexible and not power-associative, power-associative and not alternative, power-associative and not flexible
$[x,y,z] = (xy)z - x(yz)$	The associator, a trilinear function
alternative	Left and right alternative: $x(xy) = x^2y$ and $(yx)x = yx^2$
flexible	$x(yx) = (xy)x$, equivalently $[x,y,x] = 0$
power-associative	The subalgebra generated by one element is associative
$x^2x = xx^2$	Third-power associativity, the weakest of the identities of the ladder
Moufang identities	$(xyx)z = x(y(xz))$, $z(xyx) = ((zx)y)x$, $(xy)(zx) = x(yz)x$
$\mathbb{O}$	The octonion algebra, eight-dimensional over $k$, with basis $1, e_1, \ldots, e_7$
$[e_1,e_2,e_4] = 2e_7$	The witness that $\mathbb{O}$ is not associative
$x \bullet y = \tfrac{1}{2}(xy + yx)$	The symmetrised product, whose theory is Jordan Algebras', later in this Part
$\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$	The four rungs of the doubling ladder, of dimensions 1, 2, 4, 8

Non-Associative Algebras and the Property Ladder

Introduction

Algebras, the Associator and Linearisation

The Ladder of Identities

The Octonions

The Ladder as the Doubling Proceeds

What the Ladder Does Not Decide

The Neighbouring Theories

Summary

Summary of Notation

Further Reading