The Graded Action on a Module over a Complex Vector Space

Introduction

Let $V$ be a complex vector space with a Hermitian form $h$ and let $T$ be a unitary self-adjoint involution of $V$; the involution grades the space, $V = V_0\oplus V_1$, and its conjugation grades the endomorphism algebra, $E = \operatorname{End}_{\mathbb C}(V) = E_0\oplus E_1$. A module $M$ over $E$ is graded when it too is graded, $M = M_0\oplus M_1$, and the action respects the two gradings, $$ E_i\,M_j \subseteq M_{i+j} \qquad (i, j \in \{0,1\}); $$ the action is then the module-level form of the sign carried by the signed operators, an element of $E_0$ preserving the two parts of the module and an element of $E_1$ exchanging them, and the compatibility is the statement that the action commutes with the two grade involutions up to the parity sign. The grading has a geometric reading: because $T$ is unitary, the two parts $V_0, V_1$ are orthogonal and the unitary elements of $E$ that preserve the grading — the centraliser of $T$ in the unitary group — act on the graded module by isometries, and the odd unitary elements are the isometric isomorphisms that exchange the two parts. The compatibility carries no sign in the action itself; the sign rule that the article records is the Koszul sign of the graded structure, $(-1)^{|m||n|}$ wherever two odd objects are exchanged, and it is the sign of Superalgebras and Graded Structures and The Exterior Algebra, named here and not defined.

The article has three sections: the graded modules and the compatible action; the canonical example and the induced involution; and the sign rule and the Hermitian reading. Involutions of a Graded Linear Space supplies the grading of $V$ and of $E$, the grade involution and the parity of endomorphisms; The Signed Sandwich on a Complex Vector Space and The Signed Left Multiplication on a Complex Vector Space supply the signed operators; Modules over an Involutive Ring supplies the module theory over a ring with involution; and Superalgebras and Graded Structures owns the sign rule and the general theory of graded modules over graded algebras, which is named here and not used. The grading by a unitary involution and the Hermitian form are Involutive Linear Spaces, The Involution on a Complex Vector Space and Hermitian Geometry and the Unitary Group.

Throughout, $V$ is a finite-dimensional complex vector space with a positive-definite Hermitian form $h$, $T$ is a unitary self-adjoint involution, $V = V_0\oplus V_1$ is its grading with grade involution $\alpha$, $\alpha(X) = TXT$ on $E = \operatorname{End}_{\mathbb C}(V) = E_0\oplus E_1$, $M = M_0\oplus M_1$ is a graded left $E$-module with grade involution $\beta$ equal to $+\mathrm{id}$ on $M_0$ and $-\mathrm{id}$ on $M_1$, and $\rho_X$ is the action of $X \in E$ on $M$.

Graded Modules and the Compatible Action

Definition. The action of the graded algebra $E$ on the graded module $M$ is compatible with the gradings when $$ E_i\,M_j \subseteq M_{i+j} \qquad (i, j \in \{0,1\}), $$ that is when every even endomorphism preserves the two parts of $M$ and every odd endomorphism carries $M_0$ to $M_1$ and $M_1$ to $M_0$. A graded left $E$-module with a compatible action is a graded module over $E$.

Proposition (the equivalent form). The action is compatible if and only if for every homogeneous $X \in E_i$ and every $m \in M$, $$ X\,\beta(m) = (-1)^{i}\,\beta(X\,m), $$ equivalently $\beta(Xm) = (-1)^i X\beta(m)$; the two grade involutions therefore satisfy $\beta\circ\rho_X = (-1)^i\,\rho_X\circ\beta$ on the homogeneous part $E_i$.

Proof. If $m \in M_j$ and $X \in E_i$, then $Xm \in M_{i+j}$, so $\beta(Xm) = (-1)^{i+j}Xm = (-1)^iX((-1)^jm) = (-1)^iX\beta(m)$; conversely the identity for all homogeneous $m$ forces $X(M_j)\subseteq M_{i+j}$, since for $m \in M_j$ one has $\beta(Xm) = (-1)^{i+j}Xm$, which is the homogeneity statement.

Proposition (the Hermitian reading). The grading involution $T$ is unitary, so the two parts of $V$ are orthogonal, $h(V_0,V_1) = 0$, and $V_i = \ker(T \mp \mathrm{id})$. The even part $E_0$ and the odd part $E_1$ of the endomorphism algebra are the centraliser and the anticentraliser of $T$ up to the $T$-conjugation, $E_0 = \{X : XT = TX\}$, $E_1 = \{X : XT = -TX\}$; a unitary element $X$ lying in $E_0$ is an isometry of $V$ preserving the grading, and a unitary element lying in $E_1$ is an isometry exchanging the two parts.

Proof. Orthogonality of the eigenspaces of a self-adjoint involution is the computation of Reflections as Signed Two-Sided Operators on a Complex Vector Space; $X \in E_0$ means $\alpha(X) = X$, that is $TXT = X$, i.e., $XT = TX$, and $X \in E_1$ means $TXT = -X$, i.e., $XT = -TX$. A unitary $X$ is an isometry, and its parity decides whether it preserves or exchanges the $T$-eigenspaces, which are the graded parts.

Corollary (the module as a graded module over a graded algebra). The pair $(M, \beta)$ with the compatible action is a graded module over the graded algebra $(E, \alpha)$; the even part $E_0$ acts on each $M_j$ and the odd part $E_1$ maps $M_0$ to $M_1$ and $M_1$ to $M_0$, and the action is determined by the two even restrictions $E_0\times M_j\to M_j$ and the two odd restrictions $E_1\times M_j\to M_{1-j}$.

Proof. The inclusions $E_iM_j\subseteq M_{i+j}$ and the associativity of the action are the axioms of a graded module; the description of the parts is the definition of the inclusions.

The Canonical Example and the Induced Involution

Proposition (the space as its own graded module). $M = V$ with the grading $V = V_0\oplus V_1$, the grade involution $\beta = \alpha$ with $\alpha(v) = Tv$, and the evaluation action $X\cdot v = Xv$ is a graded $E$-module: the inclusions $E_iV_j\subseteq V_{i+j}$ hold because an even endomorphism preserves the two parts and an odd one exchanges them.

Proof. The action is associative and unital by the definition of composition, and the inclusions are the parity statement: $X \in E_i$ satisfies $X(V_j)\subseteq V_{i+j}$ because $XT = (-1)^iTX$ and the $V_j$ are the $T$-eigenspaces. This is Involutions of a Graded Linear Space.

Proposition (the induced involution on the module endomorphisms). The algebra $\operatorname{End}_E(M)$ of module endomorphisms is graded by $$ \operatorname{End}_E(M)_i = \{f : f(M_j)\subseteq M_{i+j}\}, $$ with grade involution $\gamma(f) = \beta f \beta$; the same holds for the algebra $\operatorname{End}_{\mathbb C}(M)$ of all complex-linear endomorphisms of $M$ when $M$ is a complex vector space.

Proof. The composition of endomorphisms adds parities, and conjugation by $\beta$ is an automorphism of order two with the two eigenspaces $f(M_j)\subseteq M_j$ and $f(M_j)\subseteq M_{1-j}$. This is the computation of Involutions of a Graded Linear Space for the endomorphism algebra of a graded space.

Remark (the Hermitian module). When $M$ carries a Hermitian form for which $\beta$ is unitary and self-adjoint, the two graded parts of $M$ are orthogonal, and the induced involution $\gamma$ on the module endomorphisms is again the conjugation by a unitary self-adjoint involution; the graded module endomorphisms of even parity are then the isometries preserving the grading and those of odd parity the isometric isomorphisms exchanging the two parts. The grading of a module is thus a parity attached to a unitary symmetry, and the compatible action is the action of the graded algebra on the graded Hermitian module.

The Sign Rule

Definition. The sign rule of the graded structure is the Koszul sign $(-1)^{|m||n|}$: it is the factor by which the flip of two homogeneous elements is corrected, $$ \tau(m\otimes n) = (-1)^{|m||n|}\,n\otimes m , $$ and it is the factor in the graded Leibniz rule of a homogeneous derivation of parity $|D|$, $$ D(mn) = (Dm)\,n + (-1)^{|D||m|}\,m\,(Dn) . $$

Proposition (the sign rule is the parity of the action). Let $X \in E_i$ act on the graded module $M$ by $\rho_X$, and suppose $M$ is an algebra on which $E$ acts by derivations. Then $X$ acts as a graded derivation of parity $i$, $$ \rho_X(mn) = \rho_X(m)\,n + (-1)^{i|m|}\,m\,\rho_X(n) , $$ and the sign is the Koszul sign of the two odd objects $X$ and $m$.

Proof. The compatibility $\beta\rho_X = (-1)^i\rho_X\beta$ says that the action of $X$ is even or odd according to $i$; an action of parity $i$ that is a derivation of the algebra $M$ is a graded derivation of parity $i$, which is exactly the displayed rule. The rule is not an extra axiom of the graded action; it is the parity of the graded structure, and its own theory is Superalgebras and Graded Structures.

Remark (the deferred sign rule). The compatibility $E_iM_j\subseteq M_{i+j}$ carries no sign: the grade involutions commute or anticommute on the homogeneous parts, and no sign enters the action itself. The sign rule that makes the tensor product of two graded modules a graded module with the Koszul flip belongs to Superalgebras and Graded Structures and to The Exterior Algebra, where the graded-commutative product $\alpha\wedge\beta = (-1)^{|\alpha||\beta|}\beta\wedge\alpha$ realises it; it is named here because the graded actions of the later articles refer to it, and it is not defined or used.

Example (the exterior algebra of a Hermitian space). Let $G$ be the unitary group of $V$ and $M = \Lambda V$ the exterior algebra, graded by the parity of the degree; the action of $G$ extends to $\Lambda V$ and preserves each degree, so it is a graded action, and the Koszul sign is the sign of the wedge product. The fixed submodule of the even subgroup — the special unitary group, which preserves the degree — is the span of the volume element in each degree where an invariant form exists, and this is the geometric instance of the graded action of the category.

Summary

A grading of a complex vector space by a unitary self-adjoint involution $T$ makes $V = V_0\oplus V_1$ an orthogonal decomposition and grades the endomorphism algebra $E = \operatorname{End}_{\mathbb C}(V) = E_0\oplus E_1$ with $E_0$ the centraliser and $E_1$ the anticentraliser of $T$; a graded module $M = M_0\oplus M_1$ over $E$ carries a compatible action exactly when $E_iM_j\subseteq M_{i+j}$, equivalently when the two grade involutions satisfy $\beta\rho_X = (-1)^i\rho_X\beta$ on the homogeneous part $E_i$, and then $E_0$ preserves the two parts of $M$ and $E_1$ exchanges them, the action being determined by the two even restrictions and the two odd ones. The canonical example is the space itself, $M = V$ with $\beta = \alpha$, $v\mapsto Tv$; the module endomorphisms are graded by $\operatorname{End}_E(M)_i = \{f : f(M_j)\subseteq M_{i+j}\}$ with grade involution $\gamma(f) = \beta f\beta$, and the unitary even and odd elements act as isometries preserving and exchanging the graded parts. The sign rule is the Koszul sign $(-1)^{|m||n|}$, appearing in the flip $\tau(m\otimes n) = (-1)^{|m||n|}n\otimes m$ and in the graded Leibniz rule $D(mn) = (Dm)n + (-1)^{|D||m|}m(Dn)$; it is the parity of the graded structure and not an independent axiom, and its theory is Superalgebras and Graded Structures and The Exterior Algebra. The signed operators are The Signed Sandwich on a Complex Vector Space and The Signed Left Multiplication on a Complex Vector Space; the adjoint action on the endomorphism module is the group - * Operator Theory.

Summary of Notation

Symbol	Meaning
$V=V_0\oplus V_1$, $T$, $\alpha$	the graded Hermitian space, its unitary involution and grade involution
$E=\operatorname{End}_{\mathbb C}(V)=E_0\oplus E_1$	the graded endomorphism algebra
$M=M_0\oplus M_1$, $\beta$	the graded module and its grade involution
$E_iM_j\subseteq M_{i+j}$	the compatibility of the action
$\beta\rho_X=(-1)^i\rho_X\beta$ on $E_i$	the equivalent form
$\operatorname{End}_E(M)_i$, $\gamma(f)=\beta f\beta$	the graded module endomorphisms and their involution
$\tau(m\otimes n)=(-1)^{\|m\|\|n\|}n\otimes m$	the Koszul flip; the sign rule