Spin Representations and Clifford Modules with Inner Conjugation

Introduction

A Clifford module is a module over the Clifford algebra of a quadratic form, and the spin representations are the Clifford modules of a special kind: the irreducible ones, and their parity-graded refinements. The internal structure theory of modules — submodules, simple and semisimple modules, Schur's lemma, the density theorem, the structure of modules over a simple algebra — is used throughout and is not repeated here; it is the theory of Modules over an Algebra and Simple and Semisimple Modules. The purpose of this article is to supply the one thing that the general theory cannot: the explicit spinor module, its construction from the quadratic form, the way the Clifford multiplication acts on it, and the decomposition into chiral halves.

The Clifford algebra $\mathrm{Cl}(V,q)$, its parity grading $\mathrm{Cl}(V,q)=\mathrm{Cl}^0\oplus\mathrm{Cl}^1$ and the graded tensor product are taken as given from the opening of the Clifford layer; only the minimum needed to state the module theory is recalled in the next paragraph, and nothing that belongs to that layer is re-derived. The base is a field $F$ of characteristic not $2$, and $q$ is non-degenerate of dimension $n$; for the definite real forms $F=\mathbb{R}$ and $q=\operatorname{diag}(+1^p,-1^q)$ the notation is $\mathrm{Cl}_{p,q}$. Complexification is written $\mathbb{C}\mathrm{l}_n=\mathrm{Cl}(V,q)\otimes_{\mathbb{R}}\mathbb{C}$ when $F=\mathbb{R}$.

Given. The Clifford algebra is generated by $V$ subject to $v^2=q(v)1$; its even part $\mathrm{Cl}^0$ is the subalgebra spanned by products of an even number of vectors, its odd part $\mathrm{Cl}^1$ by products of an odd number, and the grading satisfies $\mathrm{Cl}^i\mathrm{Cl}^j\subseteq\mathrm{Cl}^{i+j}$ for $i,j\in\mathbb{Z}/2$. The map $v\mapsto -v$ on $V$ extends to the grade involution $\alpha$, with $\alpha(x)=(-1)^kx$ on the degree-$k$ part. For the graded tensor product, $\mathrm{Cl}(V_1,q_1)\hat\otimes\mathrm{Cl}(V_2,q_2)\cong\mathrm{Cl}(V_1\oplus V_2,q_1\perp q_2)$.

Clifford Modules

Definition. A Clifford module for $(V,q)$ is a left module $S$ over the algebra $\mathrm{Cl}(V,q)$. A graded Clifford module is such a module together with a direct sum decomposition $S=S^0\oplus S^1$ such that $\mathrm{Cl}^i\cdot S^j\subseteq S^{i+j}$; the elements of $S^0$ are even and those of $S^1$ odd.

Every module carries a Clifford action of the vectors: the composite $V\hookrightarrow\mathrm{Cl}(V,q)\to\operatorname{End}_F(S)$ satisfies

$$ c(v)^2=q(v)\cdot\mathrm{id}_S, \qquad c(v)c(w)+c(w)c(v)=2g(v,w)\cdot\mathrm{id}_S, $$

which is the module form of the fundamental relation. Conversely, any assignment $V\to\operatorname{End}_F(S)$ satisfying these relations extends to a representation of $\mathrm{Cl}(V,q)$ by the universal property. The two descriptions of a Clifford module — an algebra module and a family of operators with the anticommutation relations — are equivalent, and both are used below.

Example. The regular module is $\mathrm{Cl}(V,q)$ acting on itself by left multiplication. It has dimension $2^n$ and decomposes as $\mathrm{Cl}^0\oplus\mathrm{Cl}^1$, so it is a graded Clifford module. It is not irreducible when the algebra is not a division algebra; it is the direct sum of the irreducible Clifford modules with multiplicities equal to their dimension over the appropriate division algebra.

Example. If $V$ is two-dimensional definite negative, then $\mathrm{Cl}_{0,2}\cong\mathbb{H}$ and the regular module is $\mathbb{H}$ acting on itself, of real dimension $4$. The irreducible module is $\mathbb{H}$ itself, of dimension $1$ over $\mathbb{H}$, and it is faithful and irreducible because $\mathbb{H}$ is a division algebra.

Remark. For a simple Clifford algebra the module theory is as simple as possible: there is exactly one simple module, every module is a direct sum of copies of it, and the module category is the category of vector spaces over the division algebra $D$ with $A\cong M_k(D)$. This is the content of the structure theorem for modules over a simple algebra, imported from the module theory of the corpus; what the Clifford setting adds is the identification of $D$ and $k$ from the quadratic form, which is the classification of the previous articles.

The Spinor Module from a Maximal Isotropic Subspace

The irreducible module can be written down from the form itself. The construction is due to Chevalley and proceeds through a maximal isotropic subspace.

Let $V_{\mathbb{C}}=V\otimes_F\bar F$ be the scalar extension to an algebraic closure, or suppose first that $q$ is split, so that $V$ has a maximal isotropic subspace $W$ — a subspace on which $q$ vanishes identically — of the largest possible dimension. For a non-degenerate form of dimension $n=2m$ or $2m+1$ over an algebraically closed field, a maximal isotropic subspace has dimension $m$. Choose a second maximal isotropic subspace $W'$ with $V=W\oplus W'$ and with the pairing

$$ W\times W'\longrightarrow F, \qquad (w,w')\longmapsto \beta(w,w')=q(w+w')=2g(w,w') $$

non-degenerate; the existence of such a complement is the standard splitting of a non-degenerate form.

Definition. The spinor space attached to the splitting $V=W\oplus W'$ is the exterior algebra

$$ \Delta=\Lambda^{\bullet}W=\bigoplus_{k=0}^{m}\Lambda^kW. $$

Its dimension is $2^m$.

The Clifford action is defined on the two summands separately. For $w\in W$, let $\varepsilon(w)\colon\Lambda^{\bullet}W\to\Lambda^{\bullet}W$ be exterior multiplication, $\varepsilon(w)\xi=w\wedge\xi$. For $w'\in W'$, let $\iota(w')\colon\Lambda^{\bullet}W\to\Lambda^{\bullet}W$ be the contraction characterised by the pairing, $\iota(w')(w_1\wedge\cdots\wedge w_k)=\sum_{j}(-1)^{j-1}\beta(w_j,w')w_1\wedge\cdots\widehat{w_j}\cdots\wedge w_k$; the form $\beta=2g$ rather than $g$ is the pairing that carries the Clifford relation, because $q(w+w')=2g(w,w')$ for $w,w'$ isotropic. Extend to all of $V$ by

$$ c(w+w')=\varepsilon(w)+\iota(w'). $$

Theorem. The operators $c(v)$ satisfy $c(v)^2=q(v)\cdot\mathrm{id}$ for every $v\in V$, so that $\Delta=\Lambda^{\bullet}W$ is a Clifford module of dimension $2^m$. When $n=2m$, it is the irreducible $\mathrm{Cl}(V,q)$-module over an algebraically closed field; when $n=2m+1$, it is one of the two irreducible modules, the other being $\Delta$ with the sign of the action of the volume element reversed.

Proof. For $v=w+w'$ one computes $c(v)^2=\varepsilon(w)^2+\varepsilon(w)\iota(w')+\iota(w')\varepsilon(w)+\iota(w')^2$. Exterior multiplication squares to zero, $\varepsilon(w)^2=0$, and contraction squares to zero, $\iota(w')^2=0$; the mixed terms satisfy the Clifford relation $\varepsilon(w)\iota(w')+\iota(w')\varepsilon(w)=\beta(w,w')=2g(w,w')$, the sign $(-1)^{j-1}$ in the contraction being exactly the one that pairs the $j$-th contraction slot with the $j$-th exterior slot in the expansion of the mixed terms. Hence $c(w+w')^2=2g(w,w')=q(w+w')$, the cross term $g(w,w'')$ vanishing for $w,w''\in W$ and the pairing of two elements of $W'$ vanishing as well. So the relations hold. The dimension is $2^m$ and the module is irreducible when the algebra is simple of the appropriate size; the two cases $n=2m$ and $n=2m+1$ are distinguished by the action of the volume element, computed below.

Terminology. The elements of $\Delta$ are spinors; the operator $\varepsilon$ creates a vector and the operator $\iota$ annihilates one, so that $W$ acts by creation and $W'$ by annihilation. This is the algebraic origin of the creation and annihilation operators of the exterior algebra, and it makes the spinor module the exterior algebra of a maximal isotropic subspace. The construction is the Clifford-algebra model of the exterior algebra, the two being related by the symbol of the Clifford multiplication.

Remark. Over $\mathbb{R}$ with a definite form there are no nonzero isotropic vectors, so a maximal isotropic subspace of $V$ itself does not exist. The construction is performed on the complexification $V_{\mathbb{C}}$, and the real spinor module is then obtained by a reality condition: the complex module $\Delta$ carries an antilinear structure, and the real spinors are the fixed points or the eigenspaces of that structure. This is not covered here. Over a field where the form is split — for instance over $\mathbb{C}$, or over $\mathbb{R}$ with a form of signature $(m,m)$ — the construction is already defined over the ground field.

The Complex Spin Representation

Over an algebraically closed field the classification is periodic of period two, and the spinor module is the irreducible module of the complex Clifford algebra.

Theorem. Let $n=\dim V$ and let $\mathbb{C}\mathrm{l}_n$ be the complex Clifford algebra. Then

$$ \mathbb{C}\mathrm{l}_{2m}\cong M_{2^m}(\mathbb{C}), \qquad \mathbb{C}\mathrm{l}_{2m+1}\cong M_{2^m}(\mathbb{C})\times M_{2^m}(\mathbb{C}). $$

Consequently, for $n=2m$ there is exactly one irreducible Clifford module, of complex dimension $2^m$; for $n=2m+1$ there are exactly two, each of complex dimension $2^m$. In both cases the dimension of the irreducible complex spinor module is $2^{\lfloor n/2\rfloor}$.

Proof. The isomorphisms are those of the classification over $\mathbb{C}$ (period two). The irreducible module of $M_k(\mathbb{C})$ is $\mathbb{C}^k$, of dimension $k$, and a product of two matrix algebras has one irreducible for each factor. With $k=2^m$ this gives the dimensions.

The complex spin representation. The representation of $\mathbb{C}\mathrm{l}_n$ on the irreducible module is the complex spin representation. It is the unique (or one of the two) irreducible representations of the Clifford algebra, and it restricts to the Spin group in the following section to give the spin representation proper. In the even case $n=2m$ the module $\Delta_n$ is a single irreducible, of dimension $2^m$; in the odd case $n=2m+1$ the two irreducible modules $\Delta_n^{+}$ and $\Delta_n^{-}$ are the eigenspaces of the volume element, and their direct sum is the restriction to $\mathbb{C}\mathrm{l}_n$ of the single module $\Delta_{n+1}$. The dimension of the irreducible module in either case is $2^{\lfloor n/2\rfloor}$; the two odd-case modules are not isomorphic, and they are exchanged by the volume element of $\mathbb{C}\mathrm{l}_{n+1}$, whose restriction to $\mathbb{C}\mathrm{l}_n$ is their direct sum.

Chirality and Half-Spin Representations

In even dimensions the spinor module carries an extra piece of structure: the action of the volume element splits it.

Definition. Let $\omega=e_1\cdots e_n$ be the volume element of an orthogonal basis of $V$, with $\omega^2=(-1)^{n(n-1)/2}\prod_iq(e_i)$. In the complex algebra the product $\prod_iq(e_i)$ is a square, so $\omega^2=\pm1$ and after a normalisation by a scalar one has $\omega^2=1$; the chirality operator is the resulting involution, and its $\pm1$-eigenspaces are the half-spinor spaces.

Theorem. Let $n=2m$ and let $\Delta$ be the irreducible complex Clifford module, of dimension $2^m$. Then $\omega$ acts on $\Delta$ as an involution, $\omega^2=1$, and

$$ \Delta=\Delta_+\oplus\Delta_-, \qquad \omega|_{\Delta_\pm}=\pm1, $$

with $\dim_{\mathbb{C}}\Delta_+=\dim_{\mathbb{C}}\Delta_-=2^{m-1}$. The two summands are the half-spinor modules, also called the spaces of Weyl spinors; they are interchanged by the action of $V$ (the odd part), and each is preserved by the even subalgebra $\mathrm{Cl}^0$.

Proof. In the exterior-algebra model $\Delta=\Lambda^{\bullet}W$ the volume element acts on $\Lambda^kW$ by a sign depending on the parity of $k$: the creation and annihilation operators shift the exterior degree by one, so the product of an even number of them preserves the parity of the degree and the product of $2m$ of them is a scalar on each degree, equal to $(-1)^k$ up to a normalising factor. Hence $\omega$ is diagonalisable with eigenvalues $\pm1$ and the eigenspaces are the even and the odd exterior powers, of dimensions $2^{m-1}$ each. The parity statement follows because the odd elements of the Clifford algebra (in particular the vectors) have degree one and shift $\Delta_\pm$ into $\Delta_\mp$, while the even elements preserve each summand.

Convention. For $n=2m+1$ the volume element is central, with square $\omega^2=(-1)^{m}\prod_iq(e_i)$; since $\prod_iq(e_i)$ is a square in $\mathbb{C}$, a scalar normalisation makes it an involution, and its two eigenspaces are not submodules but the two irreducible Clifford modules themselves, corresponding to a decomposition $\mathbb{C}\mathrm{l}_{2m+1}\cong\mathbb{C}\mathrm{l}_{2m}\times\mathbb{C}\mathrm{l}_{2m}$. Thus the chiral splitting happens only in even dimensions, and in odd dimensions there is instead a pair of inequivalent spin representations.

The Spin Representation of the Spin Group

The Clifford module restricts to the Spin group, because $\mathrm{Spin}(V,q)\subseteq\mathrm{Cl}^0(V,q)\subseteq\mathrm{Cl}(V,q)$ is a subgroup of the units.

Definition. Let $S$ be a Clifford module. The spin representation is the restriction of the action of $\mathrm{Cl}(V,q)$ to the Spin group,

$$ \rho\colon\mathrm{Spin}(V,q)\longrightarrow GL(S). $$

In even dimension $n=2m$ the half-spinor spaces are invariant under the spin representation, and its restrictions $\rho_\pm$ to $\Delta_\pm$ are the half-spin representations. In odd dimension the spin representation is irreducible.

Theorem. The spin representation is well defined, and $\rho(-1)=-\mathrm{id}_S$, since $S\neq0$ and $-1$ is the scalar $-1$ of the algebra. Hence $\rho$ does not descend to $SO(V,q)$: the sign of the double cover survives in the representation, and it is precisely what makes a spin representation a representation of $\mathrm{Spin}(V,q)$ rather than of the orthogonal group. If $S$ is an irreducible module over a simple $\mathrm{Cl}(V,q)$, the action of the algebra on $S$ is faithful and the kernel of $\rho$ is trivial; when $\mathrm{Cl}(V,q)\cong A\times A$ is a product, the irreducible module of one factor is annihilated by the other and the kernel of $\rho$ is the subgroup $\mathrm{Spin}(V,q)\cap(\{1\}\times A)$. A half-spin representation in even dimension can have a larger kernel, because the half-spin spaces are irreducible only over the even part: for $\mathrm{Spin}(4)\cong Sp(1)\times Sp(1)$ acting on $\Delta_+$ the kernel is $\{1\}\times Sp(1)$. Each half-spin representation in even dimension is irreducible.

Proof. The Clifford action of $\mathrm{Cl}^0$ preserves the parity grading of $S$, hence preserves $\Delta_\pm$ in the even case; the action is by algebra automorphisms and restricts to the group of units. The element $-1$ acts as $-\mathrm{id}$ on every Clifford module with $S\neq0$, so $\rho(-1)=-\mathrm{id}_S\neq\mathrm{id}_S$; since $-1$ lies in the kernel of the projection $\mathrm{Spin}(V,q)\to SO(V,q)$, the sign of the double cover survives in the representation and is exactly the obstruction to factoring through the orthogonal group. Over a simple Clifford algebra the action on an irreducible module is faithful: the annihilator of $S$ is a two-sided ideal, and it is proper because $1$ acts as $\mathrm{id}_S$, so by simplicity it vanishes and an element of the algebra acting as the identity on $S$ is the identity of the algebra; hence the kernel of $\rho$ is trivial in that case. When the algebra is a product $A\times A$, the irreducible module of $A$ is annihilated by the other factor and the kernel is the intersection $\mathrm{Spin}(V,q)\cap(\{1\}\times A)$. In even dimension the half-spin spaces are irreducible over $\mathrm{Cl}^0(V,q)$, which is the Clifford algebra of one dimension less, hence each half-spin representation is irreducible; its kernel is the kernel of the action of the even part on $\Delta_\pm$, which is larger whenever $\mathrm{Cl}^0(V,q)$ is a product of two simple factors. This happens for $\mathrm{Cl}^0_{4,0}\cong\mathbb{H}\times\mathbb{H}$, where the two factors of $\mathrm{Spin}(4)$ act on $\Delta_\pm$ by left and right multiplication by unit quaternions, so that the elements $(1,q)$ lie in the kernel of $\rho_+$ for every unit quaternion $q$.

The low-dimensional representations. In dimension three, $\mathrm{Spin}(3)\cong Sp(1)$ and the spin representation is the action of the unit quaternions on $\mathbb{H}\cong\mathbb{C}^2$ by left multiplication, of complex dimension two. In dimension four, the even subalgebra of $\mathrm{Cl}_{4,0}\cong M_2(\mathbb{H})$ is $\mathbb{H}\times\mathbb{H}$, and the two half-spin representations $\Delta_\pm$ are the two-dimensional complex representations on which the two factors of $\mathrm{Spin}(4)\cong Sp(1)\times Sp(1)$ act by left and right multiplication. These are the modules that the applications of the category use.

The Lie Algebra Representation

The spin representation of the group has a differentiated version on the orthogonal Lie algebra, and it is the form in which the representation is usually computed.

Theorem. Let $\mathrm{SO}(V,q)$ be the orthogonal Lie algebra of skew transformations. There is an isomorphism of vector spaces

$$ \Lambda^2V\longrightarrow \mathrm{SO}(V,q), \qquad v\wedge w\longmapsto \bigl(u\mapsto g(u,w)v-g(u,v)w\bigr), $$

identifying the bivectors with the skew transformations, and the Clifford algebra realises it by the commutator: the linear map

$$ \Lambda^2V\longrightarrow \mathrm{Cl}^0(V,q), \qquad v\wedge w\longmapsto \tfrac14(vw-wv), $$

is normalised so that $\bigl[\tfrac14(vw-wv),u\bigr]=g(u,w)v-g(u,v)w$ for every $u\in V$, and it is a Lie algebra homomorphism when the right-hand side is given the commutator bracket, with image the even part of degree two. Composing with the spin representation gives the infinitesimal spin representation

$$ d\rho\colon\mathrm{SO}(V,q)\longrightarrow\mathrm{GL}(S), \qquad d\rho(v\wedge w)=\tfrac14[c(v),c(w)]. $$

Proof. The identification $\Lambda^2V\cong\mathrm{SO}(V,q)$ is the standard one for a space with a non-degenerate form and is the subject of The Orthogonal Lie Algebra. For the Clifford realisation, the identity $[vw,u]=2\bigl(g(u,w)v-g(u,v)w\bigr)$, obtained from the fundamental relation by writing $vu=2g(u,v)-uv$ and $wu=2g(u,w)-uw$, gives

$$ \bigl[\tfrac14(vw-wv),u\bigr]=\tfrac12[vw,u]=g(u,w)v-g(u,v)w, $$

so the image of $v\wedge w$ is exactly the skew transformation displayed above, and the normalisation $\tfrac14$ is the one for which the two identifications agree. The element $\tfrac14(vw-wv)$ lies in $\mathrm{Cl}^0$ and spans, as $v\wedge w$ varies, the even part of degree two; writing out the commutator of two such elements in an orthogonal basis shows that it is again of degree two, so the map is a homomorphism of Lie algebras onto that space. Applying the module action gives the infinitesimal representation, $c\bigl(\tfrac14(vw-wv)\bigr)=\tfrac14[c(v),c(w)]$.

The weight decomposition. Choosing a maximal torus of $\mathrm{SO}(V,q)$ — equivalently, an orthogonal splitting of $V$ into two-dimensional planes — diagonalises the bivectors of the planes, and the infinitesimal spin representation has the weights obtained by summing the halves of the weights of the defining representation. In the exterior-algebra model the weights are the sums of the weights of the creation operators, which is why the spin representation of $\mathrm{SO}(2m)$ has highest weight $\tfrac12(\lambda_1+\cdots+\lambda_m)$ for the appropriate ordering and the half-spin representations have the two highest weights differing by the sign of the last coordinate. The computation is the standard one of the representation theory of the orthogonal group and is cited rather than repeated.

Real Spin Modules and the Eightfold Table

Over $\mathbb{R}$ the classification of the previous articles gives not one spinor module but eight kinds, according to the signature congruence class of the form.

Theorem. Let $(V,q)$ be a real quadratic space of dimension $n$ and signature difference $d=p-q$. The irreducible real Clifford module is a vector space over the division algebra $D\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$ determined by $d\bmod 8$, of dimension $k$ over $D$ as in the eightfold table. The module is

$d \bmod 8$	$D$	$\mathrm{Cl}_{p,q}$	module dimension over $D$
$0$	$\mathbb{R}$	$M_{2^{n/2}}(\mathbb{R})$	$2^{n/2}$
$1$	$\mathbb{R}$	$M_{2^{(n-1)/2}}(\mathbb{R})\times M_{2^{(n-1)/2}}(\mathbb{R})$	$2^{(n-1)/2}$
$2$	$\mathbb{R}$	$M_{2^{n/2}}(\mathbb{R})$	$2^{n/2}$
$3$	$\mathbb{C}$	$M_{2^{(n-1)/2}}(\mathbb{C})$	$2^{(n-1)/2}$
$4$	$\mathbb{H}$	$M_{2^{(n-2)/2}}(\mathbb{H})$	$2^{(n-2)/2}$
$5$	$\mathbb{H}$	$M_{2^{(n-3)/2}}(\mathbb{H})\times M_{2^{(n-3)/2}}(\mathbb{H})$	$2^{(n-3)/2}$
$6$	$\mathbb{H}$	$M_{2^{(n-2)/2}}(\mathbb{H})$	$2^{(n-2)/2}$
$7$	$\mathbb{C}$	$M_{2^{(n-1)/2}}(\mathbb{C})$	$2^{(n-1)/2}$

Proof. This is the module-theoretic reading of the eightfold table: the algebra $\mathrm{Cl}_{p,q}$ is $M_k(D)$ or a product of two such, and the irreducible module is $D^k$ over the division algebra. The real dimension of the module is $k\dim_{\mathbb{R}}D$, which for the quaternionic and complex cases is twice the complex dimension of the complex spinor module of the same formal dimension; for example for $d\equiv4$ the module is $\mathbb{H}^{2^{(n-2)/2}}$, of real dimension $2^{(n+2)/2}$. For $d\equiv1,5\bmod8$ the algebra is a product of two simple factors and the table lists one of the two irreducible modules, the other having the same division algebra and the same dimension; for $d\equiv3,7\bmod8$ the algebra is simple and the table lists its unique irreducible module.

Reality conditions. The complex spinor module of dimension $2^{\lfloor n/2\rfloor}$ carries a real structure, a complex structure or a quaternionic structure according to the same congruence: this is the meaning of the entry $D$ in the table. The real spinors are the fixed points of the real structure when $D=\mathbb{R}$, the complexification is irreducible when $D=\mathbb{C}$, and the quaternionic structure makes the complex spinor module a quaternionic space whose dimension over $\mathbb{H}$ is half its complex dimension when $D=\mathbb{H}$. The precise bookkeeping of the antilinear structures is not covered here; the table above records only the outcome.

The Biquaternion Spinor Module

The two-dimensional complex spinor module is especially transparent for the biquaternion algebra, and the applications of the category use it directly.

Example. Let $V$ be a three-dimensional real space with a definite positive form, so that $\mathrm{Cl}(V,q)\cong\mathrm{Cl}_{3,0}\cong M_2(\mathbb{C})\cong\mathbb{B}$. The complex spinor module is $\Delta=\mathbb{C}^2$, of dimension $2^{\lfloor3/2\rfloor}=2$, and the Clifford action is by the Pauli matrices. The algebra $\mathrm{Cl}_{3,0}$ is simple with center $\mathbb{C}$, so there is a single irreducible module up to isomorphism; the second irreducible module of the odd case appears only after complexification, where $\mathbb{C}\mathrm{l}_3=\mathrm{Cl}_{3,0}\otimes_{\mathbb{R}}\mathbb{C}\cong M_2(\mathbb{C})\times M_2(\mathbb{C})$ has one irreducible module $\mathbb{C}^2$ for each of its two factors, the two being exchanged by the conjugation of the complexification. The volume element $\omega$ is central in dimension three and acts on the complexified module by the two scalars $\pm i$, one on each factor. The spin group is $\mathrm{Spin}(3)\cong Sp(1)\subset\mathbb{H}$, acting on $\mathbb{C}^2$ through the identification $Sp(1)\cong SU(2)\subset M_2(\mathbb{C})$ by left multiplication.

Example. Let $V=\mathbb{R}^4$ with the form $\operatorname{diag}(+1,+1,+1,+1)$, so that $\mathrm{Cl}_{4,0}\cong M_2(\mathbb{H})$. The complex spinor module has dimension $2^{2}=4$ and splits into half-spinors of dimension two each. The even subalgebra is $\mathbb{H}\times\mathbb{H}$, and the two half-spin representations are the two-dimensional complex representations of the two factors of $\mathrm{Spin}(4)\cong Sp(1)\times Sp(1)$. This is the module of the two-sided quaternionic action used in the application layer.

Remark. In the applications of this category, the spinor module of $\mathbb{B}$ is the defining module $\mathbb{C}^2$ of $\mathbb{B}\cong M_2(\mathbb{C})$, and its dual is the conjugate module; the invariant bilinear forms on these modules are the forms under which the spin group is defined. The identification of the spinor module with the defining module of the biquaternion algebra is the content, and it is the point at which the spin representations of this article meet the number systems.

Summary

A Clifford module is a module over $\mathrm{Cl}(V,q)$; equivalently, it is a vector space with operators $c(v)$ satisfying $c(v)^2=q(v)$ and the anticommutation relations. The irreducible Clifford module can be constructed from the form: after choosing a maximal isotropic subspace $W$ of the complexified (or split) space and a complementary isotropic $W'$, the spinor space is the exterior algebra $\Delta=\Lambda^{\bullet}W$, of dimension $2^m$ for $n=2m$ or $2m+1$, with $W$ acting by exterior multiplication and $W'$ by contraction. The construction is Chevalley's.

Over an algebraically closed field, $\mathbb{C}\mathrm{l}_{2m}\cong M_{2^m}(\mathbb{C})$ has a unique irreducible module of dimension $2^m$, and $\mathbb{C}\mathrm{l}_{2m+1}\cong M_{2^m}(\mathbb{C})\times M_{2^m}(\mathbb{C})$ has two, each of dimension $2^m$. In even dimension the volume element acts as an involution on the spinor module and splits it into half-spinor spaces $\Delta_\pm$ of dimension $2^{m-1}$, exchanged by the vectors and preserved by the even subalgebra; in odd dimension it gives instead the two inequivalent spin modules. Restricting the module to $\mathrm{Spin}(V,q)\subset\mathrm{Cl}^0$ gives the spin representation, faithful up to the double cover and irreducible on each half in even dimension; its differential on $\mathrm{SO}(V,q)\cong\Lambda^2V$ is $d\rho(v\wedge w)=\tfrac14[c(v),c(w)]$.

Over $\mathbb{R}$ the spinor module is a module over $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ according to the signature difference $d\bmod8$, with dimension read from the eightfold table, and the reality condition that distinguishes the eight kinds is the real, complex or quaternionic structure carried by the complex spinor module. For definite positive three-space the module is $\mathbb{C}^2$ with the Pauli action and the spin group is $Sp(1)$, and for four-space it is $\mathbb{C}^4$ splitting into two half-spinors of dimension two on which $Sp(1)\times Sp(1)$ acts.

Summary of Notation

Symbol	Meaning
$q$, $g$	Quadratic form and polar bilinear form
$\mathrm{Cl}(V,q)=\mathrm{Cl}^0\oplus\mathrm{Cl}^1$	Clifford algebra and parity grading (given)
$\hat\otimes$	Graded tensor product (given)
$\alpha$	Grade involution
$S=S^0\oplus S^1$	Graded Clifford module and its parity
$c(v)\in\operatorname{End}_F(S)$	Clifford action of a vector, $c(v)^2=q(v)$
$V=W\oplus W'$	Splitting into maximal isotropic subspaces, $n=2m$ or $2m+1$
$\Delta=\Lambda^{\bullet}W$	Spinor space, dimension $2^m$
$\varepsilon(w)$	Exterior multiplication by $w\in W$
$\iota(w')$	Contraction by $w'\in W'$
$\beta(w,w')=q(w+w')=2g(w,w')$	Pairing of $W$ with $W'$ for $w,w'$ isotropic
$c(w+w')=\varepsilon(w)+\iota(w')$	Clifford action on $\Delta$
$\mathbb{C}\mathrm{l}_n$	Complex Clifford algebra, $\mathbb{C}\mathrm{l}_{2m}=M_{2^m}(\mathbb{C})$, $\mathbb{C}\mathrm{l}_{2m+1}=M_{2^m}(\mathbb{C})\times M_{2^m}(\mathbb{C})$
$\omega$	Volume element, chirality operator
$\Delta=\Delta_+\oplus\Delta_-$	Half-spinor (Weyl) splitting in even dimension
$\rho$, $\rho_\pm$	Spin representation and half-spin representations
$d\rho(v\wedge w)=\tfrac14[c(v),c(w)]$	Infinitesimal spin representation
$\mathrm{SO}(V,q)\cong\Lambda^2V$	Orthogonal Lie algebra as bivectors
$D\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$	Division algebra of the real spinor module, by $d\bmod8$
$\mathrm{Spin}(3)\cong Sp(1)$, $\mathrm{Spin}(4)\cong Sp(1)\times Sp(1)$	Low-dimensional spin groups