Operators on a Clifford Module

Introduction

A Clifford module is a real vector space $\mathcal{S}$ on which the Clifford algebra $A=\mathrm{Cl}(V,q)$ acts, and the operators of the function theory are built from two ingredients: the module action, which is the algebraic multiplication, and the differentiation of the functions that carry the module as their values. This article treats the operators generated by the two and the maps that commute with the module structure.

The setting is the one of The Dirac Operator: $V$ is a real vector space with basis $e_1,\dots,e_m$ and negative-definite form $q(\sum_ix_ie_i)=-\sum_ix_i^2$, $A=\mathrm{Cl}(V,q)$ with the relations $e_ie_j+e_je_i=-2\delta_{ij}$, and $\mathcal{S}$ is a left $A$-module. The operators fall into three classes. The multiplication operators are the images $c(a)$ of the algebra in $\mathrm{End}_\mathbb{R}(\mathcal{S})$, the representation by which the module is defined; the differential operators on the module of sections are those generated by the multiplications and the coordinate derivatives, and $D=\sum_\mu c(e_\mu)\partial_\mu$ is the distinguished first-order element of this algebra; the intertwiners are the operators on $\mathcal{S}$ that commute with the multiplication — the commutant — together with the module maps between two modules, and they are what the spinor theory uses to speak of invariants.

The algebraic facts about Clifford modules themselves — the classification by Bott periodicity, the module types, the irreducible real, complex and quaternionic spinor modules, the inner conjugation and the Hermitian structure — belong to Part II, to Spin Representations and Clifford Modules with Inner Conjugation, Spinors as Minimal Left Ideals with Inner Conjugation and The Number Systems as Clifford Algebras; they are cited. What this article adds is the operator layer: the representation as an algebra of operators, the commutant and its identification, the algebra of differential operators generated by the multiplication and the derivative, the symbol of that algebra, and the intertwiners as the kernel of the module action. The Hermitian refinement of these operators is Adjoints on a Clifford Module in the next group, and the operators that commute with the Clifford action on a spin manifold are Invariant Operators and Intertwiners with Hermitian Adjoint, cited.

The Clifford Multiplication as an Operator

The Representation

Definition. A Clifford module over $A$ is a real vector space $\mathcal{S}$ with a unital algebra homomorphism

$$ c : A \longrightarrow \mathrm{End}_\mathbb{R}(\mathcal{S}) , $$

the Clifford multiplication, so that $c$ restricted to $V$ is a linear map $\gamma=c|_V:V\to\mathrm{End}_\mathbb{R}(\mathcal{S})$ with

$$ \gamma(e_i)^2 = -1 , \qquad \gamma(e_i)\gamma(e_j)+\gamma(e_j)\gamma(e_i) = 0 \quad (i\neq j) . $$

The operators $\gamma(e_i)$ are skew for the natural quadratic structure of the module in the sense that $\gamma(e_i)^2=-1$; the sign is what makes the Clifford structure of the operator layer match the operator $D$ of the function theory.

Proposition (the multiplication extends and respects the relations). The map $c$ exists on all of $A$ by the universal property of the Clifford algebra, and it is the unique algebra map extending $\gamma$. The operators $c(e_A)$ for the monomials $e_A=e_{i_1}\cdots e_{i_k}$ with $i_1<\cdots

Proof. The universal property of $\mathrm{Cl}(V,q)$ states that any linear map $V\to B$ into an associative algebra $B$ with $\gamma(v)^2=q(v)1$ extends uniquely to a unital algebra map $A\to B$; taking $B=\mathrm{End}_\mathbb{R}(\mathcal{S})$ gives $c$. The linear independence is that of the monomials in the algebra. $\square$

Remark (the types). The decomposition of a Clifford module into irreducibles, and the real, complex or quaternionic character of an irreducible module, are decided by the algebra: $\mathrm{Cl}_{0,m}$ is a full matrix algebra over $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$, or a sum of two such, according to $m \bmod 8$, and an irreducible module is the corresponding standard module. The classification, the double cover $\mathrm{Spin}\to SO$ and the explicit spinor representations are Part II's, in Spin Representations and Clifford Modules with Inner Conjugation and The Number Systems as Clifford Algebras; this article uses only that $\mathcal{S}$ is a module and that its type is fixed.

The Commutant and the Intertwiners

Definition. For a Clifford module $\mathcal{S}$ the commutant (or intertwiner algebra) is

$$ \mathrm{End}_A(\mathcal{S}) = \{T\in\mathrm{End}_\mathbb{R}(\mathcal{S}) : T\,c(a)=c(a)\,T \ \text{for all }a\in A\} , $$

the algebra of $\mathbb{R}$-linear operators commuting with the Clifford multiplication. For two modules $\mathcal{S},\mathcal{T}$ the space of intertwiners is $\mathrm{Hom}_A(\mathcal{S},\mathcal{T})=\{T:\mathcal{S}\to\mathcal{T}\ :\ T c(a)=c(a)T\}$.

Theorem (Schur, and the types). If $\mathcal{S}$ is irreducible then $\mathrm{End}_A(\mathcal{S})$ is a real division algebra, hence one of $\mathbb{R},\mathbb{C},\mathbb{H}$ by the theorem of Frobenius; the type of the module is named accordingly. If $\mathcal{S}$ is a sum of $n$ copies of an irreducible module $S$ then $\mathrm{End}_A(\mathcal{S})=M_n(D)$ with $D=\mathrm{End}_A(S)$ the division algebra of the type, and $\mathrm{Hom}_A(\mathcal{S},\mathcal{T})=M_{n_\mathcal{S}\times n_\mathcal{T}}(D)$.

Proof. Quoted from the representation theory of finite-dimensional algebras: a nonzero intertwiner between irreducibles is an isomorphism, an endomorphism of an irreducible module is invertible or zero and so generates a division algebra, and the general statement is the Morita equivalence of the projector decomposition into irreducibles. $\square$

Example (the scalar module). For $\mathcal{S}=A$ with the left regular action, the commutant is the algebra of right multiplications, $\mathrm{End}_A(A)=A^{\mathrm{op}}$, and the maps $a\mapsto L_a$ are an anti-isomorphism $A\to\mathrm{End}_A(A)=A^{\mathrm{op}}$. This is the regular representation, and it exhibits the module $\mathcal{S}$ as the carrier of the whole algebra rather than of one representation of it: the scalar theory of Clifford Analysis is the case $\mathcal{S}=A$.

Example (the irreducible spinor module). For $m$ even and an irreducible complex spinor module of dimension $2^{m/2}$ the commutant is $\mathbb{C}$ in the complex category; the chirality grading of the module, which exposes the multiplication, is the extra structure that makes it a $\mathbb{Z}/2$-graded module. The real forms have commutant $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ by Bott periodicity, and the explicit modules and their commutants are Part II's.

Proposition (the double centraliser). For an irreducible $\mathcal{S}$ the algebra generated by the multiplication is its own double centraliser:

$$ \{T : T\,S=S\,T\ \text{for all }S\in\mathrm{End}_A(\mathcal{S})\} = c(A) . $$

Proof. Containment $c(A)\subseteq$ the double centraliser is immediate. The reverse inclusion is the density (or faithfulness) statement of the double-centraliser theorem for a finite-dimensional semisimple algebra; equivalently, both are the algebra generated by the image of the irreducible left module, and the module is faithful on its image. $\square$

The Differential Operators on the Module

The Algebra Generated by Multiplication and Differentiation

Definition. For an open set $\Omega\subseteq\mathbb{R}^{m+1}$ let $\mathcal{E}(\Omega;\mathcal{S})$ be the smooth $\mathcal{S}$-valued functions, on which $\mathrm{End}_\mathbb{R}(\mathcal{S})$ acts pointwise. The operator algebra of the module is the subalgebra of $\mathrm{End}_\mathbb{R}\bigl(\mathcal{E}(\Omega;\mathcal{S})\bigr)$ generated by the multiplication operators $c(a)$ for $a\in A$ and the coordinate derivatives $\partial_0,\dots,\partial_m$.

Theorem (the symbol and the associated graded). The operator algebra is filtered by the order, and its associated graded algebra is the algebra of functions on $\mathbb{R}^{m+1}$ with values in $A$ generated by the symbols; the principal symbol of a first-order operator $T=\sum_\mu a_\mu\partial_\mu+a_0$ is $\sigma(T)(x,\xi)=c\bigl(\sum_\mu a_\mu\xi_\mu\bigr)$, left multiplication by an algebra element. In particular the symbol of $D$ is $\sigma(D)(x,\xi)=c(\sum_\mu e_\mu\xi_\mu)$, and $D$ is elliptic because the symbol is invertible for $\xi\neq0$.

Proof. The product rule gives the order filtration and the reduction to the symbol algebra; the symbol of a composition is the product of the symbols, so the associated graded is generated by the symbols of $a$ and of $\partial_\mu$, which are $c(a)$ and $\xi_\mu$. The invertibility is that of The Dirac Operator. $\square$

The Distinguished Elements

Definition. The operator of the module is

$$ D = \sum_{\mu=0}^{m}c(e_\mu)\,\partial_\mu , $$

with conjugate $\bar D=\sum_\mu c(\bar e_\mu)\partial_\mu$, $\bar e_0=e_0$, $\bar e_i=-e_i$; and the vector part is $D_{\mathrm{sa}}=\sum_{i\ge1}c(e_i)\partial_i$.

Theorem (the factorisations). $D\bar D=\bar DD=\Delta\,1_{\mathcal{S}}$, where $\Delta$ is the scalar Laplacian acting componentwise, and $D_{\mathrm{sa}}^2=-\Delta\,1_{\mathcal{S}}$.

Proof. The product of the two first-order operators is a second-order operator whose symbol is $c(\sigma(\xi)\bar\sigma(\xi))=c(|\xi|^2)=|\xi|^2\,1_{\mathcal{S}}$ by the symbol computation, and whose lower-order terms vanish because the coefficients are constant; the resulting operator is the Laplacian. The second statement is the same computation for $\sum_{i\ge1}e_i\partial_i$, whose square is $-\sum_{i\ge1}\partial_i^2\,1_{\mathcal{S}}$. $\square$

Remark (the Clifford algebra of the plane). Every two-dimensional subspace of the generating space carries a copy of the Clifford algebra $\mathrm{Cl}_{0,2}\cong\mathbb{H}$ or of $\mathrm{Cl}_{1,1}$, according as the plane is definite or indefinite; the operators $c(e_i)\partial_i+c(e_j)\partial_j$ on that plane are the $\mathbb{C}$- or $\mathbb{H}$-linear Cauchy–Riemann operators of the function theory. The reader who wants the two-dimensional model may read Fueter Theory and Complex Analysis for the definite case; the indefinite case is Split-Complex Analysis on Subspaces.

The Intertwiners as Operators

Proposition (the commutant of the differential algebra). A linear operator $T$ on $\mathcal{E}(\Omega;\mathcal{S})$ that commutes with the multiplication operators $c(a)$ for all $a\in A$ is a matrix of scalar differential operators; it commutes with $D$ exactly when it commutes with all the coordinate derivatives, which for a constant-coefficient $T$ means that $T$ is a polynomial in the $\partial_\mu$ with coefficients in $\mathrm{End}_A(\mathcal{S})$. Such an operator is a differential intertwiner if it is also $A$-linear.

Proof. Commutation with $c(A)=$ the full matrix algebra generated by the multiplications forces $T$ to act by scalars in each module component, hence to be a matrix of scalar operators; commutation with $D$ and with the multiplications is equivalent to commutation with the whole algebra generated by multiplication and derivation, whose invariant operators are the polynomials in the derivatives with central coefficients; the $A$-linear ones have coefficients in the commutant. $\square$

Example (the invariant operators of the regular module). For $\mathcal{S}=A$ and the regular action, the intertwiners include the right multiplications $R_a$, and the operators commuting with the left action and with $D$ are the polynomials $\sum_A c_A\partial^A$ with coefficients in the opposite algebra $A^{\mathrm{op}}$, acting on the right. The Laplace operators $\Delta^j$ and the powers of $D$ are among them; the two-sided action of the algebra on itself is the reason the regular module is the natural setting for the integral formulae of Clifford Analysis.

Summary

A Clifford module is a vector space $\mathcal{S}$ with an algebra map $c:A\to \mathrm{End}_\mathbb{R}(\mathcal{S})$, the Clifford multiplication, whose restriction $\gamma$ to the generating space satisfies $\gamma(e_i)^2=-1$ and $\gamma(e_i)\gamma(e_j)+\gamma(e_j)\gamma(e_i)=0$. The operators on $\mathcal{S}$ fall into three classes. The multiplication operators $c(A)$ are the image of the algebra; the commutant $\mathrm{End}_A(\mathcal{S})$ is a division algebra $\mathbb{R},\mathbb{C}$ or $\mathbb{H}$ when $\mathcal{S}$ is irreducible (Schur and Frobenius), the matrix algebra $M_n(D)$ in general, and the regular module $\mathcal{S}=A$ has commutant $A^{\mathrm{op}}$ realised by the right multiplications; the module generated by multiplication may be read as its own double centraliser. The differential operators on the smooth sections are the algebra generated by $c(A)$ and the derivatives, filtered by order with the Clifford algebra as symbol algebra; its distinguished elements are $D=\sum_\mu c(e_\mu)\partial_\mu$, with $D\bar D=\bar DD=\Delta\,1_{\mathcal{S}}$, and the self-adjoint vector part $D_{\mathrm{sa}}=\sum_{i\ge1}c(e_i)\partial_i$, with $D_{\mathrm{sa}}^2=-\Delta\,1_{\mathcal{S}}$. The intertwiners are the operators commuting with the multiplication; those that also commute with $D$ are the polynomials in the derivatives with coefficients in the commutant. The classification of the modules, the spinor representations and the inner conjugation are Part II's; the adjoint of these operators with respect to the Hermitian form is Adjoints on a Clifford Module; the operators on a Clifford module over a manifold, and the elliptic complexes they define, are Clifford Modules and the Twisted Cauchy–Riemann Operator; the function theory is The Dirac Operator and Clifford Analysis.

Summary of Notation

Symbol Meaning
$A=\mathrm{Cl}(V,q)$ Clifford algebra, $e_ie_j+e_je_i=-2\delta_{ij}$
$\mathcal{S}$ Left Clifford module; $A$ itself in the scalar case
$c:A\to\mathrm{End}_\mathbb{R}(\mathcal{S})$, $\gamma=c|_V$ Clifford multiplication; $\gamma(e_i)^2=-1$
$\mathrm{End}_A(\mathcal{S})$ Commutant, the intertwiner algebra
$\mathrm{Hom}_A(\mathcal{S},\mathcal{T})$ Space of intertwiners
$D=\sum_\mu c(e_\mu)\partial_\mu$ The operator on the module; $D\bar D=\Delta\,1_\mathcal{S}$
$D_{\mathrm{sa}}=\sum_{i\ge1}c(e_i)\partial_i$ Self-adjoint vector part; $D_{\mathrm{sa}}^2=-\Delta\,1_\mathcal{S}$
$\sigma(T)(x,\xi)=c(\sum a_\mu\xi_\mu)$ Principal symbol of a first-order operator
$\mathcal{S}=A$, $\mathrm{End}_A(A)=A^{\mathrm{op}}$ The regular module and its commutant

Further Reading

  • H. Blaine Lawson and Marie-Louise Michelsohn, Spin Geometry (Princeton University Press, 1989), for Clifford modules, their types and the spinor representations.
  • R. Delanghe, F. Sommen and V. Souček, Clifford Algebra and Spinor-Valued Functions (Kluwer, 1992), for Clifford modules over function spaces and the operators on them.
  • John E. Gilbert and Margaret A. M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis (Cambridge University Press, 1991), for the operator algebra generated by the multiplication and the derivative.
  • F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis (Pitman, 1982), for the regular module and the operators with values in the algebra.