Quotients of the Tensor Algebra

Introduction

The tensor algebra $T(V)$ of Tensor Powers and the Free Algebra is the free associative algebra on the module $V$, and its universal property says that every linear map $V \to A$ into a unital associative algebra extends uniquely to an algebra homomorphism $T(V) \to A$. An algebra presented by generators and relations is obtained by imposing relations, that is, by dividing $T(V)$ by the two-sided ideal they generate. This article develops that general construction: which ideals of $T(V)$ are available, what universal property the quotient satisfies, how the grading behaves, and what the construction produces on the number systems.

The construction is general. The symmetric algebra $\operatorname{Sym}(V)$, the exterior algebra $\Lambda(V)$ and the Clifford algebra $\mathrm{Cl}(V,q)$ are three particular quotients of $T(V)$, by relations of degree two, and they are named here only to locate them within the general framework; their theory — bases, universal properties, determinants, quadratic forms — belongs to categories 06, 07 and 14 and is not developed in this article. What is fixed here is the mechanism of quotienting, not any one quotient.

Throughout, $R$ is a commutative ring with identity $1 \neq 0$ and $V$ is an $R$-module. The ideal theory of Ideals and Quotients of Algebras is used throughout, and the tensor algebra is assumed from Tensor Powers and the Free Algebra.

Ideals of the Tensor Algebra

The tensor algebra is graded, $T(V) = \bigoplus_{n \geq 0} T^n(V)$, and this grading interacts well with ideals.

Definition. A two-sided ideal $I \subseteq T(V)$ is homogeneous (or graded) if

$$ I = \bigoplus_{n \geq 0} (I \cap T^n(V)), $$

that is, if $I$ contains the homogeneous components of each of its elements. Equivalently, $I$ is generated as a two-sided ideal by homogeneous elements.

Proposition. Let $S \subseteq T(V)$ be a set of homogeneous elements and let $I = (S)$ be the two-sided ideal it generates. Then $I$ is homogeneous, equal to the direct sum of the ideals $I_n = (S_n)$ generated by the elements of $S$ of degree $n$ inside $T^n(V)$; more precisely

$$ I = \bigoplus_{n \geq 0} I_n, \qquad I_n = (S \cap T^n(V)) \cdot T(V) \cap T^n(V). $$

Proof. The ideal generated by $S$ is spanned by elements $a s b$ with $s \in S$ and $a, b$ homogeneous (since $T(V)$ is spanned by homogeneous elements and the general element is a sum of such). If $s$ is homogeneous of degree $d$ and $a, b$ have degrees $p, q$, then $a s b$ is homogeneous of degree $p + d + q$; hence every generator of $I$ is homogeneous, and a sum of homogeneous elements of possibly different degrees lies in the sum of the graded pieces it meets. Thus $I$ is the sum of its intersections with the graded pieces, and the direct sum is a sum of submodules.

Proposition (graded quotient). Let $I \subseteq T(V)$ be a homogeneous ideal. Then the quotient algebra $T(V)/I$ inherits a grading,

$$ T(V)/I = \bigoplus_{n \geq 0} T^n(V)/(I \cap T^n(V)), $$

and the product respects it: the class of a homogeneous element of degree $m$ times the class of a homogeneous element of degree $n$ is homogeneous of degree $m+n$.

Proof. The quotient of a graded module by a graded submodule is graded, with the stated graded pieces. If $x \in T^m(V)$ and $y \in T^n(V)$, then $xy \in T^{m+n}(V)$, and adding elements of $I$ changes each class within its own graded piece; the product is therefore well defined on homogeneous classes with the stated degree.

When $I$ is not homogeneous the quotient is not naturally graded; it still carries a filtration by the degree of a representative, and this is the situation of the Weyl algebra below.

Remark (the ideal generated by relations). Since $T(V)$ is associative, the two-sided ideal generated by a subset $S$ is, by Ideals and Quotients of Algebras,

$$ (S) = \Bigl\{ \sum_i a_i s_i b_i : a_i, b_i \in T(V), \ s_i \in S \Bigr\}, $$

with finitely many nonzero terms. If $S$ is empty then $(S) = 0$ and the quotient is $T(V)$.

Algebras Presented by Generators and Relations

Definition. Let $X$ be a set and let $\{r_j\}_{j \in J}$ be a family of elements of the free algebra $R\langle X\rangle$. The algebra presented by the generators $X$ and the relations $r_j = 0$ is the quotient

$$ A = R\langle X\rangle / (r_j : j \in J), $$

where $(r_j : j \in J)$ is the two-sided ideal generated by the relations.

Equivalently, in the tensor-algebra formulation, let $V$ be an $R$-module, let $T(V)$ be its tensor algebra, and let $I \subseteq T(V)$ be a two-sided ideal; the quotient is $A = T(V)/I$, and the pair $(V, I)$ is a presentation of $A$. The two formulations agree: $R\langle X\rangle = T(R^{(X)})$, and an ideal of $T(V)$ is generated by its elements, which are the relations.

Example (the number systems). The presentations collected in Ideals and Quotients of Algebras are of this form:

$$ \mathbb{C} = \mathbb{R}\langle x\rangle/(x^2+1), \qquad \mathbb{D} = \mathbb{R}\langle x\rangle/(x^2-1), \qquad \mathbb{D}' = \mathbb{R}\langle x\rangle/(x^2), $$

$$ \mathbb{H} = \mathbb{R}\langle x, y\rangle/(x^2+1,\; y^2+1,\; xy+yx). $$

In each case the relations are of degree two, and the monomials reduce: in $\mathbb{D}$, the relation $x^2 = 1$ lets every word in $x$ be rewritten as $a + bx$; in $\mathbb{H}$, the three relations rewrite every word in $x,y$ as a linear combination of the four monomials $1, x, y, xy$.

Example (a non-homogeneous presentation). Let $k$ be a field and let

$$ A_1 = k\langle x, y\rangle/(yx - xy - 1). $$

This is the Weyl algebra, the algebra of polynomial differential operators in one variable, with $x$ acting by multiplication and $y$ by differentiation. The relation is not homogeneous, because the two sides have different degrees; nevertheless $A_1$ carries a filtration in which $x$ and $y$ have filtered degree $1$, and the associated graded algebra is the commutative polynomial algebra $k[x,y]$. The Poincaré–Birkhoff–Witt theorem is the general statement that a filtered algebra presented by relations of the form $xy - yx = [x,y]$ has the symmetric algebra as its associated graded; the details belong to category 06.

The ideal generated by a subspace of $V^{\otimes 2}$. The most important case imposes relations only in degree two. Let $R_2 \subseteq V^{\otimes 2}$ be a submodule, let $I = (R_2)$ be the two-sided ideal it generates, and let

$$ A = T(V)/(R_2). $$

Then $A$ is a quadratic algebra: it is generated by $V$ and presented by relations in degree two. The quotient is graded by the proposition above, because $I$ is homogeneous when $R_2$ consists of degree-two elements. The symmetric and exterior algebras belong to this class; the Clifford algebra below imposes in addition a degree-zero term and is not quadratic in the strict sense.

The Universal Property of a Quotient

Theorem. Let $I \subseteq T(V)$ be a two-sided ideal and let $A = T(V)/I$. For every unital associative $R$-algebra $B$, precomposition with the quotient map $\pi : T(V) \to A$ is a bijection

$$ \operatorname{Hom}_{R\text{-alg}}(A, B) \;\longrightarrow\; \Bigl\{ \varphi \in \operatorname{Hom}_R(V, B) : \tilde{\varphi}(I) = 0 \Bigr\}, $$

where $\tilde{\varphi} : T(V) \to B$ is the algebra homomorphism extending $\varphi$ by the universal property of $T(V)$.

Proof. By the universal property of a quotient (Ideals and Quotients of Algebras), algebra homomorphisms $A \to B$ correspond bijectively to algebra homomorphisms $T(V) \to B$ vanishing on $I$, by $\psi \mapsto \psi \circ \pi$. By the universal property of $T(V)$, those correspond bijectively to linear maps $\varphi : V \to B$, by $\psi \mapsto \psi|_V$, with $\tilde\varphi$ the extension. Composing the two bijections gives the statement.

The theorem is the working form of the construction: to define an algebra homomorphism out of a presented algebra, it suffices to give the images of the generators and to check that the relations go to zero.

Corollary (the quotient is generated by $V$). The images of the elements of $V$ under $\pi$ generate $A = T(V)/I$ as an $R$-algebra, and the image of $T^n(V)$ generates the $n$-th filtered piece.

Corollary (presentations by generators). In the presentation $A = R\langle X\rangle/(r_j)$, an algebra homomorphism $A \to B$ is the same as a map $X \to B$ whose induced homomorphism $R\langle X\rangle \to B$ kills every relation $r_j$.

The General Low-Degree Quotients

Among the quotients by relations of degree at most two, three are singled out by a further structure on $V$, and they are the ones the nearby categories treat. They are recorded here only as quotients.

The symmetric algebra. The symmetric algebra is the quotient of $T(V)$ by the two-sided ideal generated by the commutators,

$$ \operatorname{Sym}(V) = T(V)\big/\bigl(v \otimes w - w \otimes v : v, w \in V\bigr). $$

It is the free commutative algebra on $V$, its relations are the quadratic relations $v \otimes w = w \otimes v$, the ideal is homogeneous, and the quotient is graded. For $V = R^n$ it is the polynomial algebra $R[x_1, \dots, x_n]$. The symmetric powers, the divided powers and the Jordan algebras built from this quotient belong to category 06.

The exterior algebra. The exterior algebra is the quotient of $T(V)$ by the two-sided ideal generated by the squares,

$$ \Lambda(V) = T(V)\big/\bigl(v \otimes v : v \in V\bigr). $$

It is graded, and the grading is compatible with the product; the induced product is alternating, and the quotient is finite-dimensional when $V$ is free of finite rank. Exterior powers, the determinant and the exterior calculus belong to category 07.

The Clifford algebra. Let $q : V \to R$ be a quadratic form and let $b(v,w) = q(v+w) - q(v) - q(w)$ be its associated bilinear form. The Clifford algebra is

$$ \mathrm{Cl}(V, q) = T(V)\big/\bigl(v \otimes v - q(v)\cdot 1 : v \in V\bigr). $$

The relations have components in degrees two and zero, so the ideal is not homogeneous for the $\mathbb{N}$-grading, but it is homogeneous for the $\mathbb{Z}/2$-grading of $T(V)$ by the parity of the degree; hence $\mathrm{Cl}(V,q)$ is a $\mathbb{Z}/2$-graded algebra, generated by $V$ subject to $v^2 = q(v)$. Polarising the relation gives the equivalent pair of identities

$$ v \otimes w + w \otimes v = b(v,w)\cdot 1, \qquad v \otimes v = q(v)\cdot 1 \qquad (v, w \in V), $$

so the symmetric part of the product in degree two is fixed by $b$ and only the alternating part is free; the two families are equivalent over any $R$ in which $2$ is invertible. Its theory belongs to category 14.

The quadratic form $q$ and the Clifford algebra it defines are objects of Part II, treated in Clifford Algebras; the construction is named here only because it is the third quotient of $T(V)$, and nothing in this part is proved with it.

The universal enveloping algebra. For a Lie algebra $\mathrm{G}$ with bracket $[\cdot,\cdot]$, the universal enveloping algebra is

$$ U(\mathrm{G}) = T(\mathrm{G})\big/\bigl(x \otimes y - y \otimes x - [x,y] : x, y \in \mathrm{G}\bigr), $$

the quotient of the tensor algebra of the underlying module by the relations that impose the bracket. The relations are not homogeneous when the bracket is nonzero, so $U(\mathrm{G})$ carries a filtration rather than a grading; the Poincaré–Birkhoff–Witt theorem describes it. This quotient belongs to the Lie theory of category 07.

The four examples show the reach of one construction: a choice of module $V$ and a choice of ideal $I$ generated in low degree. Nothing in the construction depends on which quotient is taken; only the theory of each quotient differs, and that theory is the business of the neighbouring categories.

Grading, Reduction and Dimension

Reduction to a normal form. Let $I \subseteq T(V)$ be an ideal generated by homogeneous relations of degree two. Then every element of $T(V)/I$ is a linear combination of classes of monomials; using the relations, one tries to rewrite every word to a normal form. The quotient is finite-dimensional exactly when the rewriting terminates in finitely many normal forms. In the number systems:

$\mathbb{C} = \mathbb{R}\langle x\rangle/(x^2+1)$ has normal forms $1, x$, so it is two-dimensional;
$\mathbb{D} = \mathbb{R}\langle x\rangle/(x^2-1)$ has normal forms $1, x$, so it is two-dimensional;
$\mathbb{D}' = \mathbb{R}\langle x\rangle/(x^2)$ has normal forms $1, x$, so it is two-dimensional;
$\mathbb{H} = \mathbb{R}\langle x,y\rangle/(x^2+1, y^2+1, xy+yx)$ has normal forms $1, x, y, xy$, so it is four-dimensional.

In each case a word of length at least two contains either a square, which the relation replaces by a scalar, or an adjacent pair $yx = -xy$ in $\mathbb{H}$, which the commutation relation reverses; iterating shortens the word. The four-dimensionality of $\mathbb{H}$ is thus a computation inside the quotient, not an extra hypothesis.

Growth of the quotient. The tensor algebra $T(V)$ has degree-$n$ rank $d^n$ when $V$ is free of rank $d$; imposing relations cuts this down. If one imposes no relation in degree $n$ then the degree-$n$ part retains rank $d^n$; if one imposes enough relations the quotient can be finite-dimensional, as in the four examples. The general invariant is the Hilbert series of the graded quotient,

$$ H_A(t) = \sum_{n \geq 0} \dim_R A_n \, t^n, $$

which is a rational function when $A$ is presented by finitely many relations of bounded degree. For $\operatorname{Sym}(V)$ with $\dim V = d$ it is $1/(1-t)^d$, for $\Lambda(V)$ it is $(1+t)^d$, and for the free algebra it is $1/(1-dt)$; the number system quotients above have polynomial Hilbert series because they are finite-dimensional.

Base change. If $R \to S$ is a homomorphism of commutative rings, then $T(V \otimes_R S) \cong T(V) \otimes_R S$ and an ideal $I \subseteq T(V)$ generates an ideal $I_S$ in $T(V) \otimes_R S$; the quotient base-changes,

$$ \bigl(T(V)/I\bigr) \otimes_R S \;\cong\; T(V \otimes_R S)/I_S. $$

This is the algebra-level form of extension of scalars, and it is what makes the complexification of a real presented algebra again a presented algebra, with the same relations.

Summary

An ideal $I$ of the graded algebra $T(V)$ is homogeneous when it is the sum of its intersections with the graded pieces; an ideal generated by homogeneous elements is homogeneous, and the quotient by a homogeneous ideal is again graded, with graded pieces $T^n(V)/(I \cap T^n(V))$. An algebra presented by generators and relations is a quotient $T(V)/I$, and its universal property is that algebra homomorphisms out of $T(V)/I$ are in bijection with linear maps $V \to B$ whose extended homomorphism kills $I$. The number systems arise this way with relations in degree two: $\mathbb{C} = \mathbb{R}\langle x\rangle/(x^2+1)$, $\mathbb{D} = \mathbb{R}\langle x\rangle/(x^2-1)$, $\mathbb{D}' = \mathbb{R}\langle x\rangle/(x^2)$ and $\mathbb{H} = \mathbb{R}\langle x,y\rangle/(x^2+1, y^2+1, xy+yx)$, of dimensions two, two, two and four.

The symmetric algebra $\operatorname{Sym}(V) = T(V)/(v\otimes w - w\otimes v)$, the exterior algebra $\Lambda(V) = T(V)/(v\otimes v)$ and the Clifford algebra $\mathrm{Cl}(V,q) = T(V)/(v\otimes v - q(v))$ are the three classical quotients by relations of low degree that carry a further structure, and the universal enveloping algebra $U(\mathrm{G}) = T(\mathrm{G})/(x\otimes y - y\otimes x - [x,y])$ is the quotient by the relations of a Lie bracket; the four are quoted here as instances of the general construction and are developed in categories 06, 07 and 14. Relations that are not homogeneous, such as $yx - xy - 1$ defining the Weyl algebra, produce a filtration rather than a grading.

Summary of Notation

Symbol	Meaning
$R$	Commutative ring with identity $1 \neq 0$
$V$	$R$-module of generators
$T(V) = \bigoplus_n T^n(V)$	Tensor algebra, graded
$I$	Two-sided ideal of $T(V)$
$T(V)/I$	Quotient algebra
$R\langle X\rangle/(r_j)$	Presentation by generators and relations
$R_2 \subseteq V^{\otimes 2}$	Space of quadratic relations
$H_A(t)$	Hilbert series of a graded algebra
$\operatorname{Sym}(V)$	Symmetric algebra, $T(V)/(v\otimes w - w\otimes v)$
$\Lambda(V)$	Exterior algebra, $T(V)/(v\otimes v)$
$\mathrm{Cl}(V,q)$	Clifford algebra, $T(V)/(v\otimes v - q(v))$
$U(\mathrm{G})$	Universal enveloping algebra of a Lie algebra
$\mathbb{C}, \mathbb{D}, \mathbb{D}', \mathbb{H}$	Number systems as low-degree quotients