The Graded Action on a Module over a Bimodule over an Algebra

Introduction

A $\mathbb{Z}/2$-graded algebra carries two structures at once: a decomposition into an even and an odd part, and the automorphism that is $+1$ on the even part and $-1$ on the odd part. The action of the algebra on a graded module is required to respect the decomposition, and respecting it forces a sign rule: moving an odd scalar past an odd element costs a sign. This article states the compatibility condition, $A_iM_j \subseteq M_{i+j}$, derives the sign rule from it, and shows how the rule turns a graded left module over a graded-commutative algebra into a graded bimodule, where the two-sided operators of The Signed Sandwich on a Bimodule over an Algebra acquire a parity.

The article is the graded member of the - Operator Theory group of this category. It uses the one-sided operators of Left and Right Multiplication of a Module and the signed operators of The Signed Sandwich on a Bimodule over an Algebra and The Signed Left Multiplication on a Module over an Algebra; the grading itself, superalgebras and their morphisms belong to Superalgebras and Graded Structures, named here and not developed. The adjoint action of the algebra on its endomorphisms is the subject of The Graded Adjoint Action on a Module over a Bimodule over an Algebra, the last article of this group. The article stays inside Part I: no distance, norm, form or limit occurs. Throughout, $R$ is a commutative ring with $1 \neq 0$ in which $2$ is invertible, $A=A_{\bar0}\oplus A_{\bar1}$ is a graded unital associative $R$-algebra, $\alpha$ is its grade involution, and $M=M_{\bar0}\oplus M_{\bar1}$ is a graded left $A$-module.

Graded Algebras, Modules and the Grade Involution

The grading and its sign operator

Definition. A graded $R$-algebra is an $R$-algebra $A$ with a direct sum decomposition $A=A_{\bar0}\oplus A_{\bar1}$ such that

$$ A_iA_j \subseteq A_{i+j} \qquad (i,j \in \mathbb{Z}/2). $$

A graded left $A$-module is a left $A$-module $M$ with a decomposition $M=M_{\bar0}\oplus M_{\bar1}$ such that

$$ A_iM_j \subseteq M_{i+j} \qquad (i,j \in \mathbb{Z}/2). $$

Elements of $A_{\bar0}$ and $M_{\bar0}$ are even, those of $A_{\bar1}$ and $M_{\bar1}$ odd, and the degree $\deg x \in \mathbb{Z}/2$ is defined for homogeneous $x$.

Definition. The grade involution of $A$ is the $R$-linear map

$$ \alpha : A \to A, \qquad \alpha(a)=(-1)^{i}a \quad (a \in A_i), $$

and the grade involution of $M$ is the analogous map on $M$.

Proposition. The grade involution of $A$ is an $R$-algebra automorphism with $\alpha^{2}=\mathrm{id}$, and its fixed part is $A_{\bar0}$; the grade involution of $M$ is additive with $\alpha^{2}=\mathrm{id}$ and satisfies

$$ \alpha(am)=\alpha(a)\,\alpha(m) \qquad (a \in A,\ m \in M). $$

Conversely, a map $\alpha$ with these properties determines the grading by $A_i=\{a : \alpha(a)=(-1)^{i}a\}$ and $M_j=\{m : \alpha(m)=(-1)^{j}m\}$.

Proof. $\alpha$ is linear and $\alpha(ab)=(-1)^{i+j}ab=\alpha(a)\alpha(b)$ for $a \in A_i$, $b \in A_j$, so it is an automorphism; $\alpha^{2}=\mathrm{id}$ because $(-1)^{2i}=1$. For the compatibility, $a \in A_i$ and $m \in M_j$ have $am \in M_{i+j}$, and both sides of $\alpha(am)=\alpha(a)\alpha(m)$ equal $(-1)^{i+j}am$. The converse is the eigenspace decomposition of an involution when $2$ is invertible. $\square$

Thus a graded module over a graded algebra is the same thing as a module with a compatible grade involution, which is the structure used by the two signed articles cited above; the grading and the involution are two descriptions of one datum.

The action is even

Proposition. The action homomorphism $\rho : A \to \operatorname{End}_R(M)$ carries $A_i$ into the operators of parity $i$:

$$ a \in A_i \implies L_a(M_j) \subseteq M_{i+j}. $$

Proof. This is the compatibility $A_iM_j \subseteq M_{i+j}$ read as a statement about the image of the left multiplication. $\square$

The action of a graded algebra on a graded module is therefore even: it preserves the degree, $\deg(am)=\deg a+\deg m$ for homogeneous $a$ and $m$.

The Sign Rule

The Koszul sign

Definition. The Koszul sign of two homogeneous elements $s$ and $t$ is $(-1)^{\deg s\,\deg t}$. It is $+1$ unless both degrees are odd, and then it is $-1$.

The grading forces the sign wherever two homogeneous things are exchanged. The first place it appears is the compatibility of the one-sided operators with the grade involution.

Proposition (the sign rule for the operators). For homogeneous $a \in A$,

$$ \alpha\,L_a\,\alpha^{-1}=(-1)^{\deg a}\,L_a, \qquad \alpha\,L_a=L_{\alpha(a)}\,\alpha . $$

More generally, for every homogeneous operator $T \in \operatorname{End}_R(M)$ of parity $\deg T$,

$$ \alpha\,T\,\alpha^{-1}=(-1)^{\deg T}\,T . $$

Proof. $\alpha L_a\alpha^{-1}(m)=\alpha(a\,\alpha^{-1}(m))=\alpha(a)m=(-1)^{\deg a}am=(-1)^{\deg a}L_a(m)$. For a homogeneous $T$ of parity $i$ and $m \in M_j$, $T(m) \in M_{i+j}$ and $\alpha T\alpha^{-1}(m)=(-1)^{j}\alpha(T(m))=(-1)^{j}(-1)^{i+j}T(m)=(-1)^{i}T(m)$. $\square$

This is the exact sense in which the grade involution imposes a sign on the operators: conjugation by $\alpha$ multiplies an operator of parity $i$ by $(-1)^{i}$, so the two-sided operators of the signed sandwich are exactly the operators twisted by that sign.

From a graded left module to a graded bimodule

A graded left module carries a canonical right action, and the sign rule is what makes it associative.

Theorem. Let $A$ be graded-commutative, that is

$$ bc=(-1)^{\deg b\,\deg c}\,cb \qquad (b,c \text{ homogeneous}), $$

and define, for homogeneous $b \in A$,

$$ R_b(m)=(-1)^{\deg b\,\deg m}\,bm, \qquad m\text{ homogeneous}. $$

Then $R$ is a right action, $R_{bc}=R_cR_b$, and the left and right actions commute:

$$ (m\cdot b)\cdot c=m\cdot(bc), \qquad a\,(m\cdot b)=(am)\cdot b . $$

Consequently $M$ is a graded $(A,A)$-bimodule, and each $R_b$ is $R$-linear of parity $\deg b$.

Proof. For the associativity, $(m\cdot b)\cdot c=(-1)^{\deg c(\deg m+\deg b)}(-1)^{\deg b\deg m}c\,b\,m$ and $m\cdot(bc)=(-1)^{(\deg b+\deg c)\deg m}(bc)m$; the two exponents differ by $\deg b\deg c$, and graded-commutativity of $A$ supplies exactly that sign, $cb=(-1)^{\deg b\deg c}bc$. For the compatibility, $a(m\cdot b)=(-1)^{\deg b\deg m}a\,b\,m$ and $(am)\cdot b=(-1)^{\deg b(\deg a+\deg m)}b\,a\,m$, whose exponents differ again by $\deg a\deg b$, supplied by graded-commutativity. The parity is read from $R_b(M_j) \subseteq M_{j+\deg b}$. $\square$

Corollary. $R_b=L_b\alpha^{\deg b}$, that is, $R_b=L_b$ for even $b$ and $R_b=L^{\alpha}_b=L_b\alpha$ for odd $b$.

Proof. For $b$ even, $R_b(m)=bm=L_b(m)$; for $b$ odd, $R_b(m)=(-1)^{\deg m}bm=b\,\alpha(m)=L_b(\alpha(m))$. $\square$

The theorem is the precise form of the assertion that the grading imposes a sign: without the sign $(-1)^{\deg b\deg m}$ the right action would not be associative, and without graded-commutativity even the signed formula fails.

The two-sided operators and their parity

Proposition. Let ${}_A M_A$ be a graded bimodule with the Koszul right action, and let $S_{a,b}=L_aR_b$ be the unsigned sandwich of The Signed Sandwich on a Bimodule over an Algebra. Then

$$ S_{a,b}(M_j) \subseteq M_{j+\deg a+\deg b}, \qquad \alpha\,S_{a,b}\,\alpha^{-1}=(-1)^{\deg a+\deg b}\,S_{a,b}, $$

and $S_{a,b}=L_{ab}\alpha^{\deg b}$. The signed sandwich $S^{\alpha}_{a,b}=S_{a,b}\alpha$ has the same parity $\deg a+\deg b$ and obeys the same sign rule.

Proof. $R_b(M_j) \subseteq M_{j+\deg b}$ and $L_a(M_{j+\deg b}) \subseteq M_{j+\deg b+\deg a}$; the sign rule is the proposition above applied to the operator of parity $\deg a+\deg b$. The identity $S_{a,b}=L_aR_b=L_aL_b\alpha^{\deg b}=L_{ab}\alpha^{\deg b}$ uses the corollary. $\square$

Since $\alpha$ has parity $0$ but acts on odd elements by $-1$, the signed sandwich $S^{\alpha}_{a,b}$ is the unsigned one with the middle factor sign-flipped on the odd part; this is the operator form of the Koszul sign, and it agrees with the composition rule of The Signed Sandwich on a Bimodule over an Algebra.

The Graded Endomorphism Algebra

The grading of the operators

Definition. The graded endomorphism algebra of $M$ is the direct sum decomposition

$$ \operatorname{End}_R(M)=\operatorname{End}_R(M)_{\bar0}\oplus\operatorname{End}_R(M)_{\bar1}, \qquad \operatorname{End}_R(M)_i=\{T : T(M_j) \subseteq M_{i+j}\}. $$

Proposition. $\operatorname{End}_R(M)$ is a graded $R$-algebra: $\operatorname{End}_i\operatorname{End}_j \subseteq \operatorname{End}_{i+j}$, the identity is even, and $L_a$ has parity $\deg a$. The grade involution acts on it by conjugation, $T\mapsto\alpha T\alpha^{-1}$, and this is an algebra automorphism of order two whose eigenvalue on $\operatorname{End}_i$ is $(-1)^{i}$.

Proof. If $S$ has parity $i$ and $T$ has parity $j$, then $ST(M_k)\subseteq S(M_{j+k})\subseteq M_{i+j+k}$, so $\operatorname{End}_i\operatorname{End}_j\subseteq\operatorname{End}_{i+j}$; the identity preserves each $M_j$. The parity of $L_a$ is the proposition on the action. The conjugation is an automorphism because $\alpha$ is invertible, and its eigenvalue is the sign rule above. $\square$

The graded commutator

The graded endomorphism algebra carries a signed commutator, and the sign rule is the statement that the action is a homomorphism of graded algebras for it.

Definition. For homogeneous operators $S,T$ the graded commutator is

$$ [S,T]=ST-(-1)^{\deg S\,\deg T}\,TS . $$

Proposition. The graded commutator is graded-alternating, $[S,T]=-(-1)^{\deg S\,\deg T}[T,S]$, it is a graded derivation in each argument,

$$ [S,TT']=[S,T]T'+(-1)^{\deg S\,\deg T}T[S,T'], $$

and an even operator $T$ is $A$-linear exactly when $[T,L_a]=0$ for all homogeneous $a$.

Proof. The alternation is a rewriting of the definition. The derivation identity is the Leibniz rule with the Koszul sign inserted before moving $T$ past $S$, checked by expanding both sides. For the last clause, an even $T$ has $\deg T=0$, so $[T,L_a]=TL_a-L_aT$, whose vanishing is $A$-linearity by Module Endomorphisms; for an odd $T$ the graded bracket carries the sign $(-1)^{\deg a}$ and its vanishing is a twisted condition, not $A$-linearity. $\square$

Examples

(a) The exterior algebra. Let $A=\Lambda(V)$ with the usual $\mathbb{Z}/2$-grading by degree parity: it is graded-commutative, so every graded left module acquires the Koszul right action and becomes a graded bimodule. The sign $(-1)^{\deg b\deg m}$ is the sign of the exterior product, and the grade involution is the parity operator.

(b) A Clifford algebra. Let $A$ be the Clifford algebra of a quadratic space, graded by the parity of the number of generators. It is not graded-commutative in general, so the Koszul right action of the theorem is available only on modules where the obstruction $bc-(-1)^{\deg b\deg c}cb$ acts trivially; the sign rule for the operators still holds, because it only uses the grade involution.

(c) The matrix algebra with an even/odd grading. For $A=M_2(k)$, $J=\operatorname{diag}(1,-1)$, $\alpha(X)=JXJ^{-1}$: the graded-commutativity is vacuous because $A$ is not graded-commutative, and the Koszul right action exists only on modules killed by $[A,A]$-type obstructions. The sign rule $\alpha T\alpha^{-1}=(-1)^{\deg T}T$ holds for every homogeneous operator.

(d) The group algebra of $\mathbb{Z}/2$. For $A=k[\mathbb{Z}/2]=k\oplus k\epsilon$ with $\epsilon$ odd and $\epsilon^{2}=1$, the algebra is graded-commutative; every graded module is a graded bimodule with $m\cdot\epsilon=(-1)^{\deg m}\epsilon m$, and $R_\epsilon=L_\epsilon\alpha$. The two-sided operators $S_{1,\epsilon}=L_\epsilon\alpha=L^{\alpha}_\epsilon$ are the signed left multiplications of The Signed Left Multiplication on a Module over an Algebra.

(e) A purely even algebra. If $A=A_{\bar0}$ then every module is graded with $M=M_{\bar0}$, the grade involution is the identity, the Koszul sign is always $+1$, and all the graded statements reduce to the ungraded ones.

Summary

A graded algebra $A=A_{\bar0}\oplus A_{\bar1}$ acts on a graded module $M=M_{\bar0}\oplus M_{\bar1}$ with the compatibility $A_iM_j\subseteq M_{i+j}$, equivalently through a grade involution $\alpha$ with $\alpha(am)=\alpha(a)\alpha(m)$; the action is even, sending $A_i$ to the operators of parity $i$. The grading imposes the Koszul sign: for homogeneous $a$, $\alpha L_a\alpha^{-1}=(-1)^{\deg a}L_a$, and for every homogeneous operator $T$, $\alpha T\alpha^{-1}=(-1)^{\deg T}T$. When $A$ is graded-commutative, the sign $R_b(m)=(-1)^{\deg b\deg m}bm$ makes any graded left module a graded bimodule, with $R_b=L_b\alpha^{\deg b}$; without graded-commutativity this right action fails to be associative. The graded endomorphism algebra $\operatorname{End}_R(M)=\operatorname{End}_{\bar0}\oplus\operatorname{End}_{\bar1}$ is a graded algebra on which the grade involution acts by conjugation with eigenvalue $(-1)^{i}$ on $\operatorname{End}_i$, and it carries the graded commutator $[S,T]=ST-(-1)^{\deg S\deg T}TS$, for which the action is a homomorphism of graded algebras. The two-sided sandwiches $S_{a,b}$ have parity $\deg a+\deg b$ and obey the same sign rule.

Summary of Notation

Symbol	Meaning
$A=A_{\bar0}\oplus A_{\bar1}$	graded $R$-algebra, $A_iA_j\subseteq A_{i+j}$
$M=M_{\bar0}\oplus M_{\bar1}$	graded left $A$-module, $A_iM_j\subseteq M_{i+j}$
$\deg x \in \mathbb{Z}/2$	degree of a homogeneous element
$\alpha$	grade involution, $+1$ on the even part, $-1$ on the odd part
$(-1)^{\deg s\deg t}$	the Koszul sign of two homogeneous elements
$L_a$, $L^{\alpha}_a$	unsigned and signed left multiplications
$R_b(m)=(-1)^{\deg b\deg m}bm$	Koszul right multiplication on a graded bimodule
$R_b=L_b\alpha^{\deg b}$	right multiplication in terms of the left one
$S_{a,b}=L_aR_b$	two-sided operator of parity $\deg a+\deg b$
$\alpha T\alpha^{-1}=(-1)^{\deg T}T$	the sign rule for a homogeneous operator
$\operatorname{End}_i$	operators shifting degree by $i$
$[S,T]=ST-(-1)^{\deg S\deg T}TS$	the graded commutator