The Graded Action on a Module over a Group

Introduction

A module over a group that carries a grading may be acted on in a way that respects the grading or in a way that twists it, and the sign rule that governs the passage between the two is the group form of the Koszul rule of a graded algebra. This article fixes the notion of a graded module over a group, states the compatibility of the action with the grading, derives the sign rule that the compatibility imposes, and identifies the graded action with the module-theoretic shadow of the signed operators of the category.

The article assumes the elementary theory of groups from Groups, the group algebra, its module theory and the augmentation from Group Algebras, and the signed left multiplication and signed sandwich from The Signed Left Multiplication on a Group and The Signed Sandwich on a Group. It uses no distance, no norm and no form; a grading is a direct sum decomposition, not a metric.

Graded Modules

Definition. Let $k$ be a commutative ring. A graded $k$-module is a $k$-module together with a direct sum decomposition

$$ M = M^{\bar0}\oplus M^{\bar1} . $$

The elements of $M^{\bar0}\cup M^{\bar1}$ are homogeneous, of degree $\bar0$ or $\bar1$; the grading involution of $M$ is the $k$-linear map $\pi_M$ equal to $+\mathrm{id}$ on $M^{\bar0}$ and $-\mathrm{id}$ on $M^{\bar1}$. A $k$-linear operator $T$ on $M$ is even if $T\pi_M=\pi_M T$ and odd if $T\pi_M=-\pi_M T$; the parity of $T$ is written $|T|\in\mathbb{Z}/2$.

Definition. A degree on a group $G$ is a homomorphism $\varepsilon : G\to\{\pm1\}$. The kernel of $\varepsilon$ is the even part $G^{\bar0}$ and its complement $G^{\bar1}=\{g:\varepsilon(g)=-1\}$ is the odd part; the collection $(G,\varepsilon)$ is a graded group.

Definition. A graded module over a graded group $(G,\varepsilon)$ is a graded $k$-module $M$ together with a $k$-linear action of $G$ such that

$$ g\cdot M^{\bar i}\subseteq M^{\overline{i+\varepsilon(g)}} \qquad\text{for all } g\in G,\ i\in\mathbb{Z}/2 . $$

An element of the even part preserves each part of $M$; an element of the odd part swaps the two parts.

Proposition (equivalence with the sign rule). The action is graded if and only if every $g$ acts by an operator of parity $\varepsilon(g)$,

$$ g\,\pi_M = \varepsilon(g)\,\pi_M\, g , $$

and then the degree of a product obeys the sign rule

$$ |g\cdot m| = \varepsilon(g) + |m| \quad\text{for } m \text{ homogeneous}. $$

Proof. On a homogeneous $m$ of degree $|m|$, the operator $\pi_M$ acts by the scalar $(-1)^{|m|}$. The condition $g\cdot M^{\bar i}\subseteq M^{\overline{i+\varepsilon(g)}}$ says that for $m$ homogeneous, $g\cdot m$ is homogeneous of degree $|m|+\varepsilon(g)$, which is the sign rule. Comparing $\pi_M g m = (-1)^{|m|+\varepsilon(g)}gm$ with $g\pi_M m=(-1)^{|m|}gm$ gives $\pi_M g=\varepsilon(g) g\pi_M$, that is $g\pi_M=\varepsilon(g)\pi_M g$. The three statements are therefore equivalent.

The parity statement is the group form of the rule $T\pi=\pm\pi T$ for an operator on a graded module; the sign $\varepsilon(g)$ plays the role of the Koszul sign attached to the degree of a homogeneous element of a graded algebra.

The Grading of the Group Algebra

The degree on the group makes the group algebra into a graded algebra, and the graded modules over the group are the graded modules over that algebra.

Proposition. If $\varepsilon : G\to\{\pm1\}$ is a degree, then the group algebra is graded by

$$ k[G]^{\bar0} = \Bigl\{\sum_{g\in G^{\bar0}} c_g\,g\Bigr\}, \qquad k[G]^{\bar1} = \Bigl\{\sum_{g\in G^{\bar1}} c_g\,g\Bigr\}, $$

and the multiplication respects the grading, $k[G]^{\bar i}\,k[G]^{\bar j}\subseteq k[G]^{\overline{i+j}}$.

Proof. The two subspaces span $k[G]$ because $G$ is the disjoint union of $G^{\bar0}$ and $G^{\bar1}$, and they meet only in $0$. The product of a term of degree $i$ and a term of degree $j$ has degree $i+j$ because $\varepsilon$ is a homomorphism, $\varepsilon(gh)=\varepsilon(g)\varepsilon(h)$.

Corollary (the compatibility of the action). A graded module over $(G,\varepsilon)$ is a graded module over the graded algebra $k[G]$, in the sense that the action is compatible with the grading,

$$ k[G]^{\bar i}\cdot M^{\bar j}\subseteq M^{\overline{i+j}} . $$

Proof. It suffices to check the inclusion on the basis $G\subseteq k[G]$, where it is the defining property of the graded action, and to extend by linearity.

Remark. The grading of the group algebra exists exactly when the degree is a homomorphism; a subset of $G$ alone does not make $k[G]$ a graded algebra. This is the group-theoretic analogue of the requirement that the grading of an algebra be compatible with its product, and it is why the degree is part of the structure $(G,\varepsilon)$ and not a property of $G$.

The Signed Action and the Koszul Sign

The graded action can be converted to one by even operators at the cost of a sign, and the cost is a cocycle.

Definition. Let $M$ be a graded module over $(G,\varepsilon)$. The signed action of $G$ on $M$ is

$$ g\triangleright m = \varepsilon(g)^{|m|}\, g\cdot m = (-1)^{\varepsilon(g)|m|}\,g\cdot m, $$

on homogeneous $m$, extended linearly.

Proposition (it is a projective action). For homogeneous $m$ the signed action satisfies

$$ g\triangleright(h\triangleright m) = (-1)^{\varepsilon(g)\varepsilon(h)}\,(gh)\triangleright m . $$

Hence the signed action is a projective action with cocycle the Koszul sign $c(g,h)=(-1)^{\varepsilon(g)\varepsilon(h)}$; it is an honest action exactly when the degree is trivial. Each operator $g\pi_M^{\varepsilon(g)}$ is even, and the assignment $g\mapsto g\pi_M^{\varepsilon(g)}$ is a projective representation with the same cocycle.

Proof. Write $\varepsilon(g)=0$ for even and $1$ for odd. The left side is $\varepsilon(h)^{|m|}\varepsilon(g)^{|h\cdot m|}g\cdot(h\cdot m)$, and $|h\cdot m|=|m|+\varepsilon(h)$, so it equals $\varepsilon(h)^{|m|}\varepsilon(g)^{|m|}\varepsilon(g)^{\varepsilon(h)}(gh)\cdot m$. The right side is $(-1)^{\varepsilon(g)\varepsilon(h)}\varepsilon(gh)^{|m|}(gh)\cdot m=(-1)^{\varepsilon(g)\varepsilon(h)}\varepsilon(g)^{|m|}\varepsilon(h)^{|m|}(gh)\cdot m$. The two agree because the remaining factor $\varepsilon(g)^{\varepsilon(h)}$ equals $(-1)^{\varepsilon(g)\varepsilon(h)}$, an identity that is trivial when $\varepsilon(h)=0$ and is $\varepsilon(g)=(-1)^{\varepsilon(g)}$ when $\varepsilon(h)=1$. The operator $g\pi_M^{\varepsilon(g)}$ is even because $\pi_M^{\varepsilon(g)}$ has parity $\varepsilon(g)$ and $g$ also has parity $\varepsilon(g)$, so the product has parity $0$; the product of two of them picks up the factor computed above, which is the cocycle.

Corollary (the two actions agree on the even part). If $g$ is even then $g\triangleright m=g\cdot m$ for all $m$; if $g$ is odd then $g\triangleright m=g\cdot m$ on $M^{\bar0}$ and $g\triangleright m=-g\cdot m$ on $M^{\bar1}$. In particular the signed action is an honest action on the even part of $G$, and the obstruction to its being an honest action on all of $G$ is the Koszul cocycle.

Relation to the Signed Operators

The graded action is the module-theoretic shadow of the signed operators of this category, in the same way that the plain action of a group on a module is the shadow of the left regular representation.

Remark. When the group carries a grade involution $\alpha$ rather than a degree homomorphism, the twisting of an action by $\alpha$, $g\cdot m\mapsto \alpha(g)\cdot m$, is the module-theoretic counterpart of the signed operators $\ell_a$ and $\Sigma^{\alpha}_{a,b}$ of The Signed Left Multiplication on a Group and The Signed Sandwich on a Group. The two structures are not the same: a degree is a homomorphism to $\{\pm1\}$, whereas a grade involution is an automorphism of order two, and a group can carry one without the other. What they share is the effect on a graded module, the insertion of a sign that distinguishes the two parts, and it is that effect that the sign rule records.

Remark (the adjoint action). The adjoint action of a graded module over a group, its compatibility with the grading and the sign rule it imposes are The Graded Adjoint Action on a Module over a Group, in the involutive part of the category; it is the case in which the twisted operator is the adjoint of the one considered here.

Summary

A graded $k$-module is a $k$-module with a decomposition $M=M^{\bar0}\oplus M^{\bar1}$, with grading involution $\pi_M$, and a degree on a group is a homomorphism $\varepsilon : G\to\{\pm1\}$. A graded module over a graded group $(G,\varepsilon)$ is a graded module with a linear action such that $g\cdot M^{\bar i}\subseteq M^{\overline{i+\varepsilon(g)}}$; equivalently each $g$ acts with parity $\varepsilon(g)$, $g\pi_M=\varepsilon(g)\pi_M g$, and the sign rule $|g\cdot m|=\varepsilon(g)+|m|$ holds on homogeneous elements. The degree makes the group algebra a graded algebra, $k[G]=k[G]^{\bar0}\oplus k[G]^{\bar1}$, and a graded module over the group is a graded module over this graded algebra, the compatibility $k[G]^{\bar i}M^{\bar j}\subseteq M^{\overline{i+j}}$ being the defining property extended by linearity. The signed action $g\triangleright m=\varepsilon(g)^{|m|}g\cdot m$ satisfies $g\triangleright(h\triangleright m)=(-1)^{\varepsilon(g)\varepsilon(h)}(gh)\triangleright m$, so it is a projective action with the Koszul cocycle $c(g,h)=(-1)^{\varepsilon(g)\varepsilon(h)}$; it agrees with the given action on the even part of $G$, differs from it by the grading involution on the odd part, and is an honest action exactly when the degree is trivial. The graded action is the module-theoretic shadow of the signed operators; a grade involution and a degree are different structures with the same effect on a graded module, the insertion of the sign that distinguishes the two parts.

Summary of Notation

Symbol	Meaning
$M=M^{\bar0}\oplus M^{\bar1}$	a graded $k$-module
$\pi_M$	the grading involution, $+\mathrm{id}$ on $M^{\bar0}$, $-\mathrm{id}$ on $M^{\bar1}$
$\varepsilon : G\to\{\pm1\}$	a degree, a homomorphism
$G^{\bar0}$, $G^{\bar1}$	even and odd parts of $G$
$g\cdot M^{\bar i}\subseteq M^{\overline{i+\varepsilon(g)}}$	the compatibility of the graded action with the grading
$g\pi_M=\varepsilon(g)\pi_M g$	the parity of the operator $g$
$\|g\cdot m\|=\varepsilon(g)+\|m\|$	the sign rule
$k[G]=k[G]^{\bar0}\oplus k[G]^{\bar1}$	the group algebra as a graded algebra
$g\triangleright m=\varepsilon(g)^{\|m\|}g\cdot m$	the signed action, a projective action
$c(g,h)=(-1)^{\varepsilon(g)\varepsilon(h)}$	the Koszul cocycle of the signed action
$\alpha$	a grade involution, the alternative twist when there is no degree