nLab Clifford algebra

Definition

Given a commutative ring $R$ and $R$-modules $M$ and $N$, an $R$-quadratic function on $M$ with values in $N$ is a map $q: M \to N$ such that the following properties hold:

• (cube relation) For any $x,y,z \in M$, we have

  $$q(x+y+z) - q(x+y) - q(x+z) - q(y+z) + q(x) + q(y) + q(z) = 0.$$

• (homogeneous of degree 2) For any $x \in M$ and any $r \in R$, we have

  $$q(r x) = r^2q(x).$$

A quadratic $R$-module is an $R$-module $M$ equipped with a quadratic form: an $R$-quadratic function on $M$ with values in $R$.

The Clifford algebra $Cl(M,q)$ of a quadratic $R$-module $(M,q)$ can be defined as the quotient of the tensor algebra $T_R(M)$ by the ideal generated by the relations $x \otimes x - q(x)$ for all $x \in M$.

Equivalently, it is the initial object in the category whose objects are pairs $(A,\phi)$ where $A$ is an associative unital $R$-algebra, and $\phi: M \to A$ is an $R$-linear map satisfying $\phi(x)^2 = q(x) 1_A$ for all $x \in M$, and whose morphisms $(A,\phi)\to (A',\phi')$ are the associative $R$-algebra maps $\chi: A\to A'$ such that $\chi\circ\phi=\phi'$.

Examples

Examples in low rank can be calculated easily. If $M$ is freely generated by a single element $e$, with quadratic form $q(e) = 1$, then $Cl(M,q) = R[e]/(e^2-1)$. Note that the opposite sign convention is often used in the differential geometry literature, so one may see the assertion that the Clifford algebra of the real line with a positive definite metric is isomorphic to the complex numbers $\mathbb{R}[e]/(e^2+1)$. Similarly, the Clifford algebra of a negative definite two dimensional real vector space is isomorphic to the (non-split) quaternions in our convention, but one may see the assertion that it is isomorphic to $M_2(\mathbb{R})$. Complexification removes the difference between positive definite and negative definite, and the two complexified algebras are isomorphic.

Let $M = L \oplus L^\vee$ for $L$ projective of rank $d$ over $R$, and $L^\vee = Hom_R(L,R)$ the dual module. One can define the canonical quadratic form $q(f+x) = f(x)$ for $f \in L^\vee$ and $x \in L$. In this case, $Cl(M,q) \cong M_{2^d}(R)$. In general, the Clifford algebra arising from a nondegenerate form is flat-locally (on $\operatorname{Spec} R$) isomorphic to a matrix algebra (when rank is even) or a direct sum of two matrix algebras (when rank is odd).

If $M$ is a projective $R$-module of rank $d$, then independently of $q$, the Clifford algebra $Cl(M,q)$ is projective of rank $2^d$, and is (noncanonically) isomorphic to $\bigwedge M$ as an $R$-module equipped with a map from $M$. The Clifford algebra is isomorphic to the exterior algebra (as algebras equipped with $R$-module maps from $M$) if and only if $q = 0$.

If $R$ is the ring of smooth functions on a pseudo-Riemannian manifold $X$, and $M$ is the $R$-module of sections of the tangent bundle, then the metric endows $M$ with a quadratic structure, and one can form the Clifford algebra of the tangent bundle.

Properties

Classification and Relation to matrix algebras

Over the complex numbers

Theorem

Let $V$ be the vector space over the complex numbers of complex dimension $d$, equipped with non-degenerate bilinear form, unique up to isomorphism. The Clifford algebra

$$Cl_{d}(\mathbb{C}) \coloneqq Cl(V)$$

is isomorphic, as a complex associative algebra to a matrix algebra as follows:

$$Cl_d(\mathbb{C}) \simeq \left\{ \array{ Mat_{2^{\tfrac{d}{2}}}(\mathbb{C}) & for \, d \, even \\ Mat_{2^{\tfrac{d-1}{2}}}(\mathbb{C}) \oplus Mat_{2^{\tfrac{d-1}{2}}}(\mathbb{C}) & for \, d \, odd } \right.$$

Remark

This is one of the incarnations of Bott periodicity.

Over the real numbers

We discuss the classification of Clifford algebras over the real numbers and their relation to matrix algebras over the real numbers. A key statement is that of the mod-8 Bott periodicity of this classification (prop. below).

In the following we write

• $\Cl_{s,t}$ for the Clifford algebra of a vector space over the real numbers with

  • $s$ generators squaring to -1

  • $t$ generators squaring to +1

• $\mathbb{C}$ for the complex numbers regarded as an associative algebra over $\mathbb{R}$;

• $\mathbb{H}$ for the quaternions regarded as an associative algebra over $\mathbb{R}$;

• $\mathbb{K}[n] \coloneqq Mat_{n \times n}(\mathbb{K})$ for the matrix algebra of $n \times n$ matrices with coefficients in $\mathbb{K}$.

Proposition

For low dimensions of real Clifford algebras, there are the following isomorphisms of associative algebras over $\mathbb{R}$

$$Cl_{0,1} \simeq \mathbb{R} \oplus \mathbb{R}$$

$$Cl_{1,1} \simeq \mathbb{R}[2]$$

$$Cl_{1,0} \simeq \mathbb{C}$$

$$Cl_{2,0} \simeq \mathbb{H}$$

(e. g. Figueroa-O’Farrill, lemma32)

Proposition

For $n,s,t \in \mathbb{N}$ then there are the following isomorphisms of associative algebras over $\mathbb{R}$:

$$Cl_{n+2,0} \simeq Cl_{0,n} \otimes_{\mathbb{R}} Cl_{2,0}$$

$$Cl_{0,n+2} \simeq Cl_{n,0} \otimes_{\mathbb{R}} Cl_{0,2}$$

$$Cl_{s+1.t+1} \simeq Cl_{s,t} \otimes_{\mathbb{R}} Cl_{1,1}$$

(e.g. Lawson-Michelsohn 89, theorem 4.1, Figueroa-O’Farrill, lemma 2)

Proposition

For $n,n_1, n_2 \in \mathbb{N}$ there are the following isomorphisms of associative algebras over $\mathbb{R}$:

• $\mathbb{R}[n_1] \otimes_{\mathbb{R}} \mathbb{R}[n_2] \simeq \mathbb{R}[n_1 n_2]$;

• $\mathbb{R}[n] \otimes_{\mathbb{R}} \mathbb{K} \simeq \mathbb{K}[n]$;

• $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \simeq \mathbb{C} \oplus \mathbb{C}$.

• $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{H} \simeq \mathbb{C}[2]$;

• $\mathbb{H} \otimes_{\mathbb{R}} \mathbb{H} \simeq \mathbb{R}[4]$

(e.g. Lawson-Michelsohn 89, proposition 4.2)

Now the incarnation in Clifford algebras of Bott periodicity over the real numbers is the following:

Proposition

For all $n \in \mathbb{N}$ there are isomorphisms of associative algebras over $\mathbb{R}$ as follows:

• $Cl_{n+8,0} \simeq Cl_{n,0} \otimes_{\mathbb{R}} Cl_{8,0}$;

• $Cl_{0,n+8} \simeq Cl_{0,n} \otimes_{\mathbb{R}} Cl_{0,8}$.

where

• $Cl_{8,0} \simeq Cl_{0,8} \simeq \mathbb{R}[16]$.

(e.g. Lawson-Michelsohn 89, theorem 4.3)

As a superalgebra

While the tensor algebra of an $R$-module $M$ has a natural integer grading, the quadratic relation collapses this to a natural $\mathbb{Z}/2\mathbb{Z}$-grading on $Cl(M,q)$.

$$Cl(M,q) = Cl(M,q)^{ev} \oplus Cl(M,q)^{odd}$$

When $M$ is projective of rank $d$, each homogeneous piece is projective of rank $2^{d-1}$. When $q$ is nondegenerate, the even part of the Clifford algebra is also flat-locally isomorphic to a matrix ring or a sum of two matrix rings.

One can view the Clifford algebra multiplication as a quantization of the commutative super algebra $\bigwedge_R M$.

Relation to orthogonal Lie algebras

Let $\{\Gamma_{a}\}_{a = 1}^n$ be the generators of an $n$-dimensional Clifford algebra over the real numbers corresponding to an inner product/metric $g$, hence with this anti-commutator:

(1)$$\big\{ \Gamma_{a} ,\; \Gamma_b \big\} \;=\; 2 g_{a b} \,.$$

Then:

Proposition

(special orthogonal Lie algebra via Clifford algebra)

The elements of the Clifford algera (1) given by

(2)$$\begin{aligned} r_{a_1 a_2} & \coloneqq\; \tfrac{1}{4} \big[ \Gamma_{a_1}, \Gamma_{a_2} \big] \\ & = \left\{ \array{ \tfrac{1}{2} \Gamma_{a_1} \Gamma_{a_2} &\vert& a_1 \neq a_2 \\ 0 &\vert& otherwise } \right. \end{aligned}$$

for all $a_1, a_2 \;\in\; \{1, \cdots, n\}$,

and equipped with the commutator bracket $[a,b] \coloneqq a b - b a$

span a Lie algebra which is isomorphic to the special orthogonal Lie algebra $\mathfrak{so}(n,g)$ with respect to $g$, in that their commutator Lie bracket is:

$$\big[ r_{a_1 a_2} \,, r_{b_1 b_2} \big] \;=\; g_{a_2 b_1} r_{a_1 b_2} - g_{a_1 b_1} r_{a_2 b_2} + g_{a_2 b_2} r_{b_1 a_1} - g_{a_1 b_2} r_{b_1 a_2}$$

Proof

First observe that

(3)$$\begin{aligned} \big[ \tfrac{1}{2} \Gamma_{a_1} \Gamma_{a_2} ,\; \Gamma_b \big] & = \tfrac{1}{2} \Gamma_{a_1} \big\{ \Gamma_{a_2}, \; \Gamma_b \big\} - \tfrac{1}{2} \big\{ \Gamma_{a_1}, \; \Gamma_b \big\} \Gamma_{a_2} \\ & = g_{a_2 b} \Gamma_{a_1} - g_{a_1 b} \Gamma_{a_2} \end{aligned}$$

Here the first step may be thought of as the graded Jacobi identity in the Clifford super Lie algebra, but it is also immediately verified by inspection. The second step evaluates the defining anti-commutators (3).

With this we compute as follows, assuming, without restriction of generality, that $a_1 \neq a_2$ and $b_1 \neq b_2$:

$$\begin{aligned} \big[ \tfrac{1}{4} [\Gamma_{a_1}, \Gamma_{a_2}] , \; \tfrac{1}{4} [\Gamma_{b_1}, \Gamma_{b_2}] \big] & = \tfrac{1}{4} \Big[ \big[ \tfrac{1}{4} [\Gamma_{a_1},\Gamma_{a_2}] ,\, \Gamma_{b_1} \big] ,\, \Gamma_{b_2} \Big] + \tfrac{1}{4} \Big[ \Gamma_{b_1}, \big[ \tfrac{1}{4} [\Gamma_{a_1},\Gamma_{a_2}] ,\, \Gamma_{b_2} \big] \Big] \\ & = g_{a_2 b_1} \tfrac{1}{4} [\Gamma_{a_1},\Gamma_{b_2}] - g_{a_1 b_1} \tfrac{1}{4} [\Gamma_{a_2},\Gamma_{b_2}] + g_{a_2 b_2} \tfrac{1}{4} [\Gamma_{b_1},\Gamma_{a_1}] - g_{a_1 b_2} \tfrac{1}{4} [\Gamma_{b_1}, \Gamma_{a_2}] \end{aligned}$$

Here the first step is again the graded Jacobi identity (and is again also immediely checked by inspection), while the second step uses (3).

Relation to $Spin$ groups

Let $M$ be a projective $R$-module of finite rank, and let $q$ be nondegenerate. Write $Cl(M,q)^\times$ for the group of units of the Clifford algebra $Cl(M,q)$.

The Clifford group $\Gamma_{M,q}(R)$ is the subgroup of elements $x$ for which twisted conjugation stabilizes the submodule $M \subset Cl(M,q)$. Here, twisted conjugation is defined by $y \mapsto x y\alpha(x)^{-1}$, where $\alpha$ is the automorphism of $CL(M,q)$ induced by the $-1$ map on $M$. Since twisted conjugation by $M$-stabilizing elements amounts to reflection $y \mapsto y - 2\frac{(x,y)}{q(x)}x$, there is a canonical map $\Gamma_{M,q}(R) \to O(M,q)$, and the Clifford group is in fact a central extension of the orthogonal group by $R^\times$.

The Clifford group is made of homogeneous elements in the $\mathbb{Z}/2\mathbb{Z}$-grading, and the subgroup of even elements is a normal subgroup of index two. One also has a spinor norm $Q: \Gamma_{M,q}(R) \to R^\times$ on the Clifford group, defined by $Q(x) = x^t x$, where $x \mapsto x^t$ is the anti-involution of the Clifford algebra defined by opposite multiplication in the tensor algebra.

The Pin group $Pin_{M,q}(R)$ is the group elements of the Clifford group with spinor norm 1. The Spin group $Spin_{M,q}(R)$ is the group of elements in the even subgroup of the Clifford group with spinor norm 1.

The restriction of the map $\Gamma_{M,q}(R) \to O(M,q)$ to the Pin group may not be surjective, but it is for positive definite real vector spaces. The kernel is the group $\mu_2(R)$ of elements of $R$ that square to 1. Similarly, the Spin group has a map to the special orthogonal group with kernel $\mu_2(R)$, but it may not be surjective in general.

One can use base change to define the groups given above as functors on commutative $R$-algebras.

Relation to exterior algebra (quantization)

For $V$ an inner product space, the symbol map (see there) constitutes an isomorphism of the underlying super vector spaces of the Clifford algebra with the exterior algebra on $V$.

One may understand the Clifford algebra as the quantization Grassmann algebra induced from the inner product regarded as an odd symplectic form.

Spinors

For nondegenerate quadratic forms on real vector spaces, spinors/spin representations are distinguished linear representations of Spin groups that are not pulled back from the corresponding special orthogonal groups. In other words, the central element $-1$ acts nontrivially. They can be realized as restrictions of representations of the even parts of Clifford algebras. Since even parts of Clifford algebras are (up to complexification) the sum of one or two matrix rings, their representation theory is quite simple.

The specific nature of spinor representations possible depends on the signature of the vector space modulo 8. This is a manifestation of Bott periodicity. One always has a Dirac spinor - the fundamental (spin) representation of the complexified Clifford algebra. In even dimensions, this splits into two Weyl spinors (called half-spin representations). One may also have real representations called Majorana spinors, and these may decompose into Majorana-Weyl spinors.

There are infinite dimensional Clifford algebra constructions that appear in conformal field theory. One may extend the above discussion to topological $R$-modules and continuous quadratic forms, and one obtains canonical central extensions of infinite dimensional groups and algebras by a relative determinant construction. Semi-infinite wedge spaces are spinor modules for Clifford algebras of quadratic Tate $R$-modules.

Warnings

There is a difference of sign convention between differential geometers (following Atiyah) and everyone else.

Clifford algebras are often defined using bilinear forms instead of quadratic forms (and one often sees incorrect definitions of quadratic forms in terms of bilinear forms). Such definitions will yield wrong (or boring) objects when 2 is not invertible.

Special orthogonal groups are often defined as the kernel of the determinant map on the corresponding orthogonal groups, but in characteristic 2, the determinant is trivial, while the Clifford grading (called the Dickson invariant) is not.

The Clifford group is sometimes defined without the twist in the conjugation, and this means the map to the orthogonal group may not be a surjection, and the action on $M$ is by negative reflections.

The spinor norm is sometimes defined with the opposite sign.

Special orthogonal groups over the reals are sometimes defined to be the connected component of the identity in the orthogonal group. In indefinite signature, this defines an index two subgroup of the special orthogonal group.

Spin groups in signature $(m,n)$ for $m,n \geq 2$ have fundamental groups of order two. They are simply connected when $m$ or $n$ is at most one.

Related concepts

• geometric algebra

• Clifford module,

• Clifford bundle

• Clifford module bundle

• Fierz identity

• spin geometry, spin group

• 2-Clifford algebra

• Feynman slash notation

• Clifford ring

• Pythagorean ring

References

The notion of Clifford algebra is due to

• William Clifford, Applications of Grassmann’s extensive algebra, American Journal of Mathematics 1 4 (1878) 350-358 [doi:10.2307/2369379, jstor:2369379]

  (referring to Hermann Grassmann‘s Ausdehnungslehre)

Further early discussion:

• Emil Artin, §V.4 in: Geometric Algebra, Wiley 1957 (1988) [ISBN:978-1-118-16454-9, Wikipedia entry, ark:/13960/t4nk37034]

• Élie Cartan, Theory of Spinors, Dover, (1966)

Brief introductions:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Ch II.7 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, chII.7: pdf]

• Jean Gallier, Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions (arXiv:0805.0311)

• Marc Lachieze-Rey, Spin and Clifford algebras, an introduction (arXiv:1007.2481)

Standard textbook accounts:

• H. Blaine Lawson, Marie-Louise Michelsohn, Spin geometry, Princeton University Press (1989) [ISBN 9780691085425]

• Pertti Lounesto, Clifford Algebras and Spinors, London Mathematical Society 286, Cambridge University Press (2001) [doi:10.1017/CBO9780511526022]

See also:

• Eckhard Meinrenken: Clifford algebras and Lie groups, Ergebn. Mathem. & Grenzgeb., Springer (2013) [doi:10.1007/978-3-642-36216-3]

For a program that promotes the use of Clifford algebra as a good expositional tool in introductory mechanics see Geometric Algebra.

See also the discussion of Majorana spinors

• José Figueroa-O'Farrill, Majorana spinors (pdf)

On subgroups of Clifford algebras reminiscent of gauge groups of relevance in the standard model of particle physics and grand unified theories thereof:

• Robert A. Wilson, A group-theorist’s perspective on symmetry groups in physics (arXiv:2009.14613)

• Robert A. Wilson, On the Problem of Choosing Subgroups of Clifford Algebras for Applications in Fundamental Physics, Adv. Appl. Clifford Algebras 31, 59 (2021) (doi:10.1007/s00006-021-01160-5)

(For similar investigations, see also here at Albert algebra.)