List of Associative Algebras

Introduction

This article lists the associative algebras that the corpus meets, together with the weaker identities — alternativity, flexibility, power-associativity and the Moufang identities — that survive the failure of associativity. Associativity is the rung on which most of the corpus is built: an algebra is associative when its associator vanishes identically, and the constructions of the category Linear Algebras are stated for the associative case. Every entry points to the article that introduces the object.

The list is a proper subclass of List of Algebras: that article carries the algebras over a field of the corpus with their dimension, their centre and their ideals, and this one carries the associative algebras, over a field or over a commutative ring, with the identity that fixes them, and beside them the algebras that fail associativity.

This article introduces nothing and proves nothing. It records examples and non-examples side by side, a non-example being an algebra whose product is not associative, with the failure named and the article that records it.

The Associative Identity and the Associator

An algebra is associative when $(xy)z = x(yz)$ for all elements, equivalently when the associator vanishes identically. The associator is trilinear, so associativity is a linear condition in each argument, and the weaker identities of the corpus ladder are obtained from it by linearisation.

Object	The property it has	Introduced in
Associative algebra	the product satisfies the polynomial identity $(xy)z - x(yz) = 0$	Associative Algebras
The associator $[x,y,z] = (xy)z - x(yz)$	the trilinear map that measures the failure of associativity; it vanishes identically exactly for the associative algebras	Non-Associative Algebras and the Property Ladder
Power-associative algebra	the subalgebra generated by one element is associative; implied by associativity, and strictly weaker	Non-Associative Algebras and the Property Ladder
Alternative algebra	the left and right laws $x(xy) = x^2y$ and $(yx)x = yx^2$; implied by associativity	Non-Associative Algebras and the Property Ladder
Flexible algebra	$x(yx) = (xy)x$, equivalently $[x,y,x] = 0$; implied by associativity	Non-Associative Algebras and the Property Ladder
The three Moufang identities	$(xyx)z = x(y(xz))$, $z(xyx) = ((zx)y)x$, $(xy)(zx) = x(yz)x$; implied by alternativity, hence by associativity	Non-Associative Algebras and the Property Ladder

The implications are one-directional and each step that the corpus separates is strict, with a witness in the article that separates it; the theorem is stated over a field of characteristic not $2$ or $3$. The octonions are alternative and not associative, so alternativity does not imply associativity; the algebra $B$ of Non-Associative Algebras and the Property Ladder is power-associative and not alternative; and the algebra $A$ of the same article is flexible and not power-associative, so flexibility and power-associativity are incomparable. Associativity is the strongest rung of the ladder, third-power associativity is the weakest identity the article records, and alternativity is the weakest classical rung at which a calculus of powers is still available, power-associativity being the statement of that calculus.

The Free Associative Algebras

The tensor algebra $T(V)$ is the free associative algebra on the module $V$, and the free algebra is the case of a free module on a set of generators. Their universal property is the reason they are the source of every algebra presented by generators and relations.

Algebra	The property it has	Introduced in
$T(V)$, the tensor algebra	the free associative $R$-algebra on the module $V$, with product concatenation	Tensor Powers and the Free Algebra
$R\langle x_1,\dots,x_n\rangle$, the free algebra	the free associative $R$-algebra on $n$ generators; the non-commuting polynomial algebra	Tensor Powers and the Free Algebra
The universal property of $T(V)$	every $R$-linear map $V \to A$ into a unital associative $R$-algebra extends uniquely to an algebra homomorphism $T(V) \to A$	Tensor Powers and the Free Algebra
$kQ$, the path algebra of a quiver	associative, with the directed paths as a basis and concatenation as the product; graded by the length of the path	Quiver Representations and Representation Type

A quotient of an associative algebra by a two-sided ideal is associative, so every algebra presented by generators and relations is associative. That single fact is why the constructions below are associative without a second verification.

The Quotients That Remain Associative

The symmetric, exterior and Clifford algebras are quotients of $T(V)$ by homogeneous relations of degree two, the Weyl algebra is the quotient by a relation of degree two that is not homogeneous, and the group algebra and the polynomial algebra are associative by their definitions.

Algebra	The property it has	Introduced in
$R[x_1,\dots,x_n]$, the polynomial algebra	commutative and associative	Polynomial Algebras
$\operatorname{Sym}(V)$, the symmetric algebra	the commutative associative quotient of $T(V)$ by the commutators	The Symmetric Algebra
$\Lambda(V)$, the exterior algebra	the graded-commutative associative quotient by $v \otimes v$	The Exterior Algebra
$\mathrm{Cl}(V,q)$, the Clifford algebra	the associative quotient by $v \otimes v - q(v)$	Clifford Algebras
$A_1(k) = k\langle x,y\rangle/(xy - yx - 1)$, the Weyl algebra	associative, a domain, and not a quotient by homogeneous relations	Non-Commutative Domains, §The Weyl Algebra
$k[G]$, the group algebra	associative; the group law extended bilinearly	Group Algebras
$\mathbb{C}, \mathbb{D}, \mathbb{D}'$ as quotients of $T(V)$	the number systems presented by generators and relations; associative	Quotients of the Tensor Algebra
$\mathbb{H}$, the quaternions	associative division algebra, and not commutative	Quaternion Algebra
$\mathbb{B} \cong M_2(\mathbb{C})$, the biquaternions	associative, with zero divisors	Biquaternion Algebra
$\mathbb{H}_{\mathbb{D}} \cong \mathbb{H} \oplus \mathbb{H}$, the split biquaternions	associative, with zero divisors	Split-Biquaternion Algebra

The group algebra is associative because the group law is, and $k[G]$ is commutative exactly when $G$ is abelian; the exterior algebra is associative but only graded-commutative, so it is not a commutative algebra for $\dim V \geq 2$. These are the standard associative non-commutative algebras of the corpus, and the boundary between them and the commutative ones is commutativity, not associativity.

The Model: Matrix and Endomorphism Algebras

The matrix algebra is the model by which the associative theory is tested, and every finite-dimensional simple associative algebra is a matrix algebra over a division ring.

Algebra	The property it has	Introduced in
$M_n(R)$, the matrix algebra	associative, with unit $I_n$ and basis the matrix units $E_{ij}$	Matrix Algebras
$\operatorname{End}_R(R^n)$	the endomorphism algebra, isomorphic to $M_n(R)$	Matrix Algebras
$\operatorname{End}_A(M)$, an endomorphism ring	associative and unital; its units are the automorphisms of $M$	Modules over an Algebra
$M_n(D)$ over a division ring $D$	associative, simple and Artinian; the general simple finite-dimensional algebra	Simple and Semisimple Modules
The trace form $\langle A,B\rangle = \operatorname{Tr}(AB)$	symmetric, associative and non-degenerate on $M_n(k)$	Matrix Algebras

The matrix algebra is the worked case of almost every statement of the associative theory: the centre, the ideals, the units and the tensor product formula are all computed there first. Its place as the model is recorded in Matrix Algebras, and the matrix rings are gathered in List of Matrix Rings.

The Universal Enveloping Algebra of a Lie Algebra

The universal enveloping algebra is the associative algebra attached to a non-associative one: it is the quotient of the tensor algebra that turns a Lie bracket into a commutator, and it is the corpus's standard passage from the non-associative to the associative side.

Object	The property it has	Introduced in
$U(\mathrm{G})$, the universal enveloping algebra	the associative $R$-algebra $T(\mathrm{G})$ modulo $x \otimes y - y \otimes x - [x,y]$	Universal Enveloping Algebras
The universal property of $U(\mathrm{G})$	every $R$-linear $\varphi : \mathrm{G} \to A$ into an associative $R$-algebra with $\varphi([x,y]) = \varphi(x)\varphi(y) - \varphi(y)\varphi(x)$ factors uniquely through $\mathrm{G} \to U(\mathrm{G})$	Universal Enveloping Algebras
The commutator algebra $A_{\mathrm{Lie}}$	the Lie algebra with bracket $[a,b] = ab - ba$ attached to an associative algebra $A$; not associative	Universal Enveloping Algebras
The adjunction	$U$ is the left adjoint of the functor attaching to an associative algebra its commutator Lie algebra $A_{\mathrm{Lie}}$	Universal Enveloping Algebras
The Poincaré–Birkhoff–Witt theorem	the ordered monomials in a basis of $\mathrm{G}$ form a basis of $U(\mathrm{G})$, and $\mathrm{gr}\,U(\mathrm{G}) \cong \operatorname{Sym}(\mathrm{G})$	Universal Enveloping Algebras

The enveloping algebra is associative by construction, and it is the reason a representation of a Lie algebra is a module over an associative algebra. The Lie algebra itself is the canonical non-example, and it is recorded in the last section with the other algebras that fail associativity.

Power-Associativity Beyond Associativity

Associativity implies power-associativity, and the converse fails; the corpus carries two algebras that separate the middle steps of the ladder, so that the strictness of every implication above is visible in the list itself.

Algebra	The property it has	Introduced in
The algebra $A$ with basis $u, v$ and $u^2 = v$, $uv = vu = v$, $v^2 = 0$	commutative, hence flexible, but not power-associative: $(u^2)(u^2) = 0$ while $((u^2)u)u = v$	Non-Associative Algebras and the Property Ladder
The algebra $B$, the symmetrised $M_2(k)$ with $x \bullet y = \tfrac{1}{2}(xy + yx)$	power-associative but not alternative, for $k$ of characteristic not $2$	Non-Associative Algebras and the Property Ladder

These two algebras are constructed in the ladder article to separate the rungs of the ladder rather than met as named algebras elsewhere, and they are included here because they show that associativity is not recovered from the weaker identities in either direction.

Algebras That Fail Associativity

The following objects are the ones a reader might expect among the associative algebras and will not find, and the objects that show the ladder is strict at the top.

Object	Why it is not associative	Introduced in
$\mathbb{O}$, the octonions	the associator is nonzero, $[e_1,e_2,e_4] = 2e_7$; alternative, flexible and power-associative	Octonion Algebra
$\mathbb{S}$, the sedenions	the doubling loses the multiplicative norm: it has zero divisors and is not a division algebra, and it is neither associative nor alternative	Division Algebras
A Lie algebra $\mathrm{G}$	the bracket is anticommutative and satisfies the Jacobi identity; associativity is replaced	Lie Algebras
A Jordan algebra	the product is commutative and satisfies the Jordan identity in place of associativity	Jordan Algebras
The cross product on $\mathbb{R}^3$	fails associativity: it is antisymmetric and satisfies the Jacobi identity	Algebras
The commutator algebra $A_{\mathrm{Lie}}$	the bracket of an associative algebra is not associative	Universal Enveloping Algebras

The octonions are the standard witness: they are the first algebra of the Cayley–Dickson chain at which associativity is lost, and they retain alternativity and power-associativity, so they separate the top rung from the next. The sedenions lose alternativity as well, and the Lie and Jordan algebras replace associativity by an identity rather than weakening it.

Warnings

Object	Why it is absent from the list	Introduced in
An algebra satisfying the Moufang identities that is not alternative	not exhibited: the corpus leaves the pair of alternativity and the Moufang identities unseparated	Non-Associative Algebras and the Property Ladder
An algebra satisfying third-power associativity that is not flexible	not exhibited: flexibility implies third-power associativity, and no algebra of the corpus separates the two	Non-Associative Algebras and the Property Ladder

Summary

This article has listed the associative algebras of the corpus. The property that gathers them is the vanishing of the associator $[x,y,z] = (xy)z - x(yz)$, a polynomial identity and a linear condition in each argument; power-associativity, alternativity, flexibility and the Moufang identities are the weaker identities implied by it, and the corpus's algebras $A$ and $B$ show that the ladder is strict. The free associative algebras are the tensor algebra $T(V)$ and the free algebra, with the universal property that makes every presented algebra a quotient; the symmetric, exterior and Clifford algebras, the Weyl algebra, the group algebra and the polynomial algebra remain associative because they are quotients of $T(V)$ or are defined by an associative product. The model is the matrix algebra $M_n(R)$ and its endomorphism algebra $\operatorname{End}_A(M)$, the simple finite-dimensional case being $M_n(D)$ over a division ring. The universal enveloping algebra $U(\mathrm{G})$ is the associative algebra attached to the Lie algebra $\mathrm{G}$ by a quotient of $T(\mathrm{G})$. The non-examples — the octonions, the sedenions, the Lie algebras, the Jordan algebras, the cross product on $\mathbb{R}^3$ and the commutator algebra $A_{\mathrm{Lie}}$ — each name the failure of associativity.

Summary of Notation

A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following.

Symbol	Meaning
$[x,y,z] = (xy)z - x(yz)$	The associator, a trilinear function vanishing exactly on the associative algebras
$T(V)$, $R\langle x_1,\dots,x_n\rangle$	The tensor algebra and the free algebra
$\operatorname{Sym}(V)$, $\Lambda(V)$, $\mathrm{Cl}(V,q)$	The symmetric, exterior and Clifford algebras
$A_1(k)$	The Weyl algebra $k\langle x,y\rangle/(yx - xy - 1)$
$k[G]$, $kQ$	The group algebra and the path algebra of a quiver
$M_n(R)$, $E_{ij}$, $I_n$	The matrix algebra, its matrix units and identity matrix
$\operatorname{End}_A(M)$, $\operatorname{End}_R(R^n)$	Endomorphism ring of a module, and of $R^n$
$M_n(D)$	The matrix ring over a division ring
$U(\mathrm{G})$, $A_{\mathrm{Lie}}$	The universal enveloping algebra, and the commutator Lie algebra of an associative algebra
$\mathbb{O}$, $\mathbb{S}$	The octonions and the sedenions
$x \bullet y = \tfrac{1}{2}(xy + yx)$	The symmetrised product