Clifford Algebras

Introduction

The Clifford algebra of a quadratic form is the associative algebra generated by the space, subject to the single relation $v^2 = q(v)\cdot 1$. It is the universal solution of that relation: every linear map from the space to an algebra that squares to the form factors through it. This universality is what makes the construction functorial, and the relation, polarised, is the anticommutation law that turns the algebra into a deformation of the exterior algebra.

This article constructs the Clifford algebra, proves the universal property and the fundamental relation, establishes the parity grading, records the functoriality that the rest of the Clifford layer uses, and defines the graded tensor product and the real signature algebras. It is the opening of the Clifford layer of the category, and it fixes the conventions — the sign in the defining relation, the parity grading, the notation $\mathrm{Cl}(V, q)$ against $\mathrm{Cl}_{p,q}$ — on which the later entries depend. The base is a commutative ring $R$ in which $2$ is invertible, and the module is free of finite rank; the quadratic form, its polar form and its radical are from Bilinear Forms and Quadratic Forms and Polarisation, and no general quadratic-form theory is restated. The tensor algebra is from category 05, and the exterior algebra from The Exterior Algebra.

Construction and Universal Property

The Tensor Algebra Quotient

Let $R$ be a commutative ring, $V$ a free $R$-module, and $q : V \to R$ a quadratic form with polar form $B$, so that $q(v) = B(v, v)$ and

$$ q(u + v) = q(u) + 2B(u, v) + q(v). $$

Let $T(V) = \bigoplus_{k \geq 0} V^{\otimes k}$ be the tensor algebra, with product the concatenation of tensors and with $V = V^{\otimes 1}$ sitting inside it.

Definition. The Clifford algebra $\mathrm{Cl}(V, q)$ is the quotient of the tensor algebra $T(V)$ by the two-sided ideal $I$ generated by the elements

$$ v \otimes v - q(v)\cdot 1, \qquad v \in V. $$

It is an associative algebra with identity over $R$, generated by the image of $V$. The image of $v \in V$ is again written $v$, so that the defining relations read

$$ v^2 = q(v)\cdot 1 \qquad (v \in V). $$

The construction makes sense over any commutative ring; when $R$ is a field, one writes $F$ in place of $R$ and speaks of the dimension rather than the rank.

The Universal Property

Theorem (universal property). Let $A$ be an associative algebra with identity over $R$, and let $f : V \to A$ be an $R$-linear map such that

$$ f(v)^2 = q(v)\cdot 1_A \qquad (v \in V). $$

Then there is a unique homomorphism of $R$-algebras $\tilde{f} : \mathrm{Cl}(V, q) \to A$ with $\tilde{f}(v) = f(v)$ for all $v \in V$.

Proof. The universal property of the tensor algebra gives a unique algebra homomorphism $T(V) \to A$ extending $f$. For $v \in V$ its value on $v \otimes v - q(v)\cdot 1$ is $f(v)^2 - q(v)\cdot 1_A = 0$, so the homomorphism kills the ideal $I$ and descends to $\mathrm{Cl}(V, q)$; uniqueness is inherited from $T(V)$.

Corollary (uniqueness). The pair $(\mathrm{Cl}(V, q), \iota)$, where $\iota : V \to \mathrm{Cl}(V, q)$ is the structure map, is unique up to a unique isomorphism: if $(\mathrm{Cl}', \iota')$ has the same universal property, then there are mutually inverse algebra homomorphisms between $\mathrm{Cl}(V, q)$ and $\mathrm{Cl}'$ carrying $\iota$ to $\iota'$.

Proof. Apply the universal property of each to the structure map of the other.

So the Clifford algebra is characterised by the universal property and not by the quotient presentation, which is only one way to realise it.

The Fundamental Relation

Polarisation

Theorem. In $\mathrm{Cl}(V, q)$ the generators satisfy

$$ uv + vu = 2B(u, v)\cdot 1 \qquad (u, v \in V). $$

Proof. Since $q(u + v) = q(u) + 2B(u, v) + q(v)$ and the square of a sum in an associative algebra expands as

$$ (u + v)^2 = u^2 + uv + vu + v^2, $$

the defining relations give

$$ q(u) + 2B(u, v) + q(v) = q(u) + uv + vu + q(v), $$

so $uv + vu = 2B(u, v)\cdot 1$.

Corollary. If $u$ and $v$ are orthogonal, $B(u, v) = 0$, then $uv = -vu$.

Proof. Put $B(u, v) = 0$ in the fundamental relation.

The relation $uv + vu = 2B(u, v)$ is called the fundamental relation of the Clifford algebra. It is the polarised form of the defining relation and is the working form of the anticommutation law: in an orthogonal basis of a non-degenerate form the generators satisfy $e_i e_j = -e_j e_i$ for $i \neq j$ and $e_i^2 = q(e_i)$.

Remark. The relation is not an independent assumption. It is a consequence of $v^2 = q(v)$ and of the definition of the polar form, and it is the reason the Clifford algebra is a deformation of the exterior algebra: setting $q = 0$ in the defining relation gives $v^2 = 0$, and the Clifford algebra becomes the exterior algebra $\Lambda(V)$ of The Exterior Algebra. The two algebras have the same underlying graded module but different products, and the deformation is measured by the filtration of the next section.

The Parity Grading

The Grading

The tensor algebra $T(V)$ is $\mathbb{Z}$-graded, with $V^{\otimes k}$ in degree $k$, and the defining relations are not homogeneous but are the sum of a term of degree $2$ and a term of degree $0$. Consequently, $\mathrm{Cl}(V, q)$ inherits only the parity of the tensor grading, and this parity survives exactly.

Definition. Let $\mathrm{Cl}^0(V, q)$ be the image in $\mathrm{Cl}(V, q)$ of the even part $\bigoplus_k V^{\otimes 2k}$ of the tensor algebra, and $\mathrm{Cl}^1(V, q)$ the image of the odd part $\bigoplus_k V^{\otimes 2k+1}$. These are the even and odd parts of the Clifford algebra, and

$$ \mathrm{Cl}(V, q) = \mathrm{Cl}^0(V, q) \oplus \mathrm{Cl}^1(V, q). $$

Proposition. The decomposition is a $\mathbb{Z}/2$-grading:

$$ \mathrm{Cl}^i(V, q)\,\mathrm{Cl}^j(V, q) \subseteq \mathrm{Cl}^{i+j}(V, q), \qquad i, j \in \mathbb{Z}/2\mathbb{Z}. $$

In particular $\mathrm{Cl}^0(V, q)$ is a subalgebra with the same identity, and $\mathrm{Cl}^1(V, q)$ is a module over it; the product of two odd elements is even.

Proof. The parity of a product of tensor monomials is the sum of their parities, so a product of an element of parity $i$ and an element of parity $j$ is a sum of monomials of parity $i + j$; this is the inclusion. The closure of $\mathrm{Cl}^0$ under multiplication is the case $i = j = 0$.

Proposition. Let $e_1, \ldots, e_n$ be an orthogonal basis of $V$, so that $B(e_i, e_j) = 0$ for $i \neq j$. Then $e_ie_j = -e_je_i$ for $i \neq j$, and a product $e_{i_1}\cdots e_{i_k}$ with distinct indices changes by the sign $(-1)^{|\sigma|}$ when the factors are reordered by a permutation $\sigma$. Consequently the associated graded algebra of the degree filtration is the exterior algebra $\Lambda(V)$.

Pro. The anticommutation is the fundamental relation with $B(e_i, e_j) = 0$. Reordering the factors multiplies by the sign of the transpositions performed, and the square of a generator is a scalar, so the products of distinct generators give a well-defined associated graded algebra; it has the defining relations $e_i^2 = 0$ and $e_ie_j = -e_je_i$ of the exterior algebra, and by the dimension count it is exactly $\Lambda(V)$.

The grading is a parity and not a degree: unlike the exterior algebra, the Clifford algebra has no $\mathbb{Z}$-grading, because $e_i^2 = q(e_i)$ is a scalar and not zero.

The Grade Involution

Definition. The grade involution $\alpha$ is the algebra automorphism of $\mathrm{Cl}(V, q)$ defined by $\alpha(x) = (-1)^k x$ for $x \in \mathrm{Cl}^k(V, q)$, $k \in \{0, 1\}$. Equivalently, $\alpha$ is the unique algebra automorphism whose restriction to $V$ is $v \mapsto -v$, which exists by the universal property applied to the map $-1_V : V \to \mathrm{Cl}(V, q)$.

Proposition. The grade involution is an involution, $\alpha^2 = \mathrm{id}$, and its eigenspaces are the even and odd parts:

$$ \mathrm{Cl}^0(V, q) = \{x : \alpha(x) = x\}, \qquad \mathrm{Cl}^1(V, q) = \{x : \alpha(x) = -x\}. $$

Proof. The map $v \mapsto -v$ squares to the same map as $v \mapsto v$ on the generators, since $(-v)^2 = v^2 = q(v)$; by the universal property it extends to an algebra endomorphism, and applying it twice is the identity on generators, hence everywhere. The eigenspace description is the definition.

Remark. The grade involution is one of the three standard involutions of the Clifford algebra. The other two, reversion and Clifford conjugation, together with the volume element and the centre, are properties of the finite-dimensional theory and are treated in Clifford Algebras in Finite Dimensions.

Functoriality

Isometries Induce Isomorphisms

Theorem. Let $f : (V, q) \to (W, q')$ be an isometry of quadratic spaces, so that $q'(f(v)) = q(v)$ for all $v$. Then there is a unique algebra isomorphism $\mathrm{Cl}(f) : \mathrm{Cl}(V, q) \to \mathrm{Cl}(W, q')$ with

$$ \mathrm{Cl}(f)\bigl(\iota_V(v)\bigr) = \iota_W\bigl(f(v)\bigr) \qquad (v \in V), $$

where $\iota_V : V \to \mathrm{Cl}(V, q)$ and $\iota_W : W \to \mathrm{Cl}(W, q')$ are the structure maps.

Proof. The composite $V \to W \to \mathrm{Cl}(W, q')$ is linear and satisfies $v \mapsto f(v)$ with $f(v)^2 = q'(f(v)) = q(v)$, so the universal property produces an algebra homomorphism $\mathrm{Cl}(V, q) \to \mathrm{Cl}(W, q')$. The same construction applied to $f^{-1}$ gives an inverse.

Corollary. The assignment $(V, q) \mapsto \mathrm{Cl}(V, q)$ is a functor from the category of quadratic spaces with isometries to the category of algebras with algebra homomorphisms, and it respects identities and composition.

Corollary. Every isometry $T$ of $(V, q)$ induces an algebra automorphism $\mathrm{Cl}(T)$ of $\mathrm{Cl}(V, q)$, and the assignment

$$ \operatorname{O}(V, q) \longrightarrow \operatorname{Aut}(\mathrm{Cl}(V, q)), \qquad T \longmapsto \mathrm{Cl}(T) $$

is a group homomorphism. Its image is the group of automorphisms of the Clifford algebra induced by isometries of the form, and it is the structural origin of the Clifford, Pin and Spin groups of the layer.

Base Change and Direct Sums

Proposition. For a ring homomorphism $R \to S$ there is a natural isomorphism

$$ \mathrm{Cl}(V, q) \otimes_R S \cong \mathrm{Cl}(V \otimes_R S,\ q \otimes_R S). $$

Proof. Both sides satisfy the same universal property over $S$: the right side universally adjoins to $V \otimes_R S$ a square root of $q \otimes_R S$, and the left side is the scalar extension of the universal algebra over $R$.

The behaviour under orthogonal direct sums is the subject of the graded tensor product.

The Graded Tensor Product

Definition

Let $A = A^0 \oplus A^1$ and $B = B^0 \oplus B^1$ be $\mathbb{Z}/2$-graded algebras over $R$.

Definition. The graded tensor product $A \,\hat{\otimes}\, B$ is the algebra whose underlying module is $A \otimes_R B$, with grading

$$ (A \,\hat{\otimes}\, B)^k = \bigoplus_{i + j = k} A^i \otimes_R B^j, \qquad i, j, k \in \mathbb{Z}/2\mathbb{Z}, $$

and with multiplication determined by

$$ (a \otimes b)(a' \otimes b') = (-1)^{|b||a'|}\,(aa') \otimes (bb'), $$

where $|b|$ and $|a'|$ are the parities of the homogeneous elements $b$ and $a'$. Thus odd elements from the two factors anticommute, while all other pairs commute.

Proposition. The graded tensor product is associative and commutative up to the natural isomorphisms of graded algebras, and it is the coproduct in the category of $\mathbb{Z}/2$-graded algebras.

Proof. The sign $(-1)^{|b||a'|}$ is the Koszul sign, and the associativity and commutativity follow from the cocycle property of the sign; the coproduct property is the universal property of the tensor product combined with the grading.

The Splitting Theorem

Theorem. Let $q_1 \perp q_2$ be the orthogonal sum of the forms $q_1$ on $V_1$ and $q_2$ on $V_2$, on the space $V_1 \oplus V_2$. Then there is a natural isomorphism of $\mathbb{Z}/2$-graded algebras

$$ \mathrm{Cl}(V_1 \oplus V_2,\ q_1 \perp q_2) \cong \mathrm{Cl}(V_1, q_1) \,\hat{\otimes}\, \mathrm{Cl}(V_2, q_2). $$

Proof. Let $A = \mathrm{Cl}(V_1, q_1)$ and $B = \mathrm{Cl}(V_2, q_2)$, and define an $R$-linear map

$$ g : V_1 \oplus V_2 \longrightarrow A \,\hat{\otimes}\, B, \qquad g(v_1, v_2) = v_1 \otimes 1 + 1 \otimes v_2. $$

Then

$$ g(v_1, v_2)^2 = v_1^2 \otimes 1 + 1 \otimes v_2^2 + (v_1 \otimes 1)(1 \otimes v_2) + (1 \otimes v_2)(v_1 \otimes 1), $$

and the two cross terms cancel because $(v_1 \otimes 1)(1 \otimes v_2) = v_1 \otimes v_2$ and $(1 \otimes v_2)(v_1 \otimes 1) = -v_1 \otimes v_2$ by the sign rule. Hence $g(v_1, v_2)^2 = (q_1(v_1) + q_2(v_2))\cdot 1$, and the universal property gives an algebra homomorphism from the left side to the right. It is an isomorphism: the restriction to each summand gives homomorphisms $\mathrm{Cl}(V_i, q_i) \to \mathrm{Cl}(V_1 \oplus V_2, q_1 \perp q_2)$ by the universal property, and these combine into a two-sided inverse because the products $v_1^a v_2^b$ with the parity rule generate the target.

Corollary. The parities multiply as

$$ \mathrm{Cl}^k(V_1 \oplus V_2,\ q_1 \perp q_2) = \bigoplus_{i + j = k} \mathrm{Cl}^i(V_1, q_1) \otimes_R \mathrm{Cl}^j(V_2, q_2), \qquad k \in \mathbb{Z}/2\mathbb{Z}. $$

In particular the even part of the tensor product is not the tensor product of the even parts: it contains the tensor product of the odd parts as well. This is exactly the statement that the splitting is graded and not ordinary, and it is why the classification of Clifford algebras is periodic rather than simply multiplicative.

The Conventions of the Category

The following conventions are fixed by this article and are used without restatement by the rest of the Clifford layer.

The Sign Convention

The defining relation is $v^2 = q(v)\cdot 1$, so that

$$ \mathrm{Cl}(V, q) \text{ is generated by } V \text{ with } v^2 = q(v)\cdot 1, \qquad uv + vu = 2B(u, v)\cdot 1. $$

The associated form is $B$ with $q(v) = B(v, v)$, never $2B$; a generator of square $-1$ therefore corresponds to $q(v) = -1$, and the Clifford form of a number system is the negative of its norm on the generators, as recorded in the companion entry with this article. The relation $v^2 = q(v)\cdot 1$ holds for every vector and not only for the elements of an orthogonal basis; in a hyperbolic plane, for instance, an isotropic basis $e, f$ with $q(e) = q(f) = 0$ and $B(e, f) = 1$ gives generators with $e^2 = f^2 = 0$ and $ef + fe = 2$.

The Parity Grading

The algebra is $\mathbb{Z}/2$-graded as $\mathrm{Cl}(V, q) = \mathrm{Cl}^0(V, q) \oplus \mathrm{Cl}^1(V, q)$, with $\mathrm{Cl}^0$ the even part and $\mathrm{Cl}^1$ the odd part; the grade involution $\alpha$ acts as $+1$ on $\mathrm{Cl}^0$ and $-1$ on $\mathrm{Cl}^1$ and on generators is $v \mapsto -v$. The superscript is a parity, not a degree; the $\mathbb{Z}$-grading survives only as a filtration, whose associated graded algebra is the exterior algebra.

$\mathrm{Cl}(V, q)$ against $\mathrm{Cl}_{p,q}$

The symbol $\mathrm{Cl}(V, q)$ denotes the Clifford algebra of an arbitrary quadratic space over a commutative ring. When $R = \mathbb{R}$ and the form is the signature form

$$ q_{p,q}(x_1, \ldots, x_n) = x_1^2 + \cdots + x_p^2 - x_{p+1}^2 - \cdots - x_n^2, \qquad n = p + q, $$

on $\mathbb{R}^n$, the algebra is written

$$ \mathrm{Cl}_{p,q} = \mathrm{Cl}(\mathbb{R}^{p+q}, q_{p,q}). $$

Thus $\mathrm{Cl}_{p,q}$ has $p$ generators of square $+1$ and $q$ generators of square $-1$, all pairwise anticommuting. The complexification of a real Clifford algebra is written

$$ \mathbb{C}\mathrm{l}_n = \mathrm{Cl}(V, q) \otimes_{\mathbb{R}} \mathbb{C}, \qquad \dim_{\mathbb{R}} V = n, $$

and depends on $n$ only, because over $\mathbb{C}$ the signs of the form can be normalised to $+1$. The classification of the algebras $\mathrm{Cl}_{p,q}$ and $\mathbb{C}\mathrm{l}_n$, and the identification of the low-dimensional cases with the number systems, belong to the other entries of the category and are not anticipated here.

Functoriality

An isometry of quadratic spaces induces an isomorphism of Clifford algebras, so the Clifford algebra is a functor on quadratic spaces with isometries; every isometry of $(V, q)$ induces an algebra automorphism, giving the group homomorphism $\operatorname{O}(V, q) \to \operatorname{Aut}(\mathrm{Cl}(V, q))$ from which the Clifford group is built.

Summary

The Clifford algebra $\mathrm{Cl}(V, q)$ of a quadratic form $q$ on a free $R$-module $V$ is the quotient of the tensor algebra $T(V)$ by the two-sided ideal generated by the elements $v \otimes v - q(v)\cdot 1$, so that $V$ generates it subject to $v^2 = q(v)\cdot 1$. It satisfies the universal property: every $R$-linear map $f : V \to A$ into an associative algebra with $f(v)^2 = q(v)\cdot 1_A$ extends uniquely to an algebra homomorphism $\mathrm{Cl}(V, q) \to A$. The universal property determines the algebra up to unique isomorphism.

The fundamental relation $uv + vu = 2B(u, v)\cdot 1$ follows from the defining relation by polarisation and requires $2$ invertible in $R$; in particular orthogonal vectors anticommute. Setting $q = 0$ recovers the exterior algebra, so the Clifford algebra is a deformation of it.

The algebra carries a parity grading $\mathrm{Cl}(V, q) = \mathrm{Cl}^0 \oplus \mathrm{Cl}^1$ with $\mathrm{Cl}^i\mathrm{Cl}^j \subseteq \mathrm{Cl}^{i+j}$; the even part is a subalgebra, the odd part a module over it, and the two odd generators anticommute up to the scalar $2B(u, v)$. The grade involution $\alpha$, induced by $v \mapsto -v$, is the algebra automorphism acting as $(-1)^k$ on $\mathrm{Cl}^k$, and its eigenspaces are the two parts.

The construction is functorial: an isometry $f : (V, q) \to (W, q')$ induces an algebra isomorphism $\mathrm{Cl}(f)$, so the orthogonal group of $q$ acts on $\mathrm{Cl}(V, q)$ by algebra automorphisms, and scalar extension commutes with the construction. Under an orthogonal direct sum the Clifford algebra is the graded tensor product, $\mathrm{Cl}(V_1\oplus V_2, q_1\perp q_2) \cong \mathrm{Cl}(V_1, q_1)\,\hat\otimes\,\mathrm{Cl}(V_2, q_2)$, where odd elements from the two factors anticommute by the Koszul sign; the parity of a decomposable element is the sum of the parities of its factors. The conventions fixed here are the sign $v^2 = q(v)$, the parity grading $\mathrm{Cl}^0 \oplus \mathrm{Cl}^1$, and the notation $\mathrm{Cl}(V, q)$ for a general quadratic space against $\mathrm{Cl}_{p,q}$ for the real signature form, with $\mathbb{C}\mathrm{l}_n$ for its complexification.

Summary of Notation

Further Reading

• I. R. Porteous, Clifford Algebras and the Classical Groups (Cambridge University Press, 1995), for the universal property, functoriality and the graded tensor product.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge University Press, 2nd ed. 2001), for the defining and fundamental relations and a computational treatment.
• H. Blaine Lawson and Marie-Louise Michelsohn, Spin Geometry (Princeton University Press, 1989), for the parity grading and the deformation of the exterior algebra.
• Max-Albert Knus, Quadratic and Hermitian Forms over Rings, Grundlehren der mathematischen Wissenschaften 294 (Springer, 1991), for the Clifford algebra over a commutative ring with $2$ invertible.
• Nicolas Bourbaki, Algebra I (Springer, 1998), for the universal property of the Clifford algebra and its functorial behaviour.