G-module

In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G. Group (co)homology provides an important set of tools for studying general G-modules.

The torus can be made an abelian group isomorphic to the product of the circle group. This abelian group is a Klein four-group-module, where the group acts by reflection in each of the coordinate directions (here depicted by red and blue arrows intersecting at the identity element).

The term G-module is also used for the more general notion of an R-module on which G acts linearly (i.e. as a group of R-module automorphisms).

Definition and basics

Let ${\displaystyle G}$ be a group. A left ${\displaystyle G}$-module consists of^[1] an abelian group ${\displaystyle M}$ together with a left group action ${\displaystyle \rho :G\times M\to M}$ such that

${\displaystyle g\cdot (a_{1}+a_{2})=g\cdot a_{1}+g\cdot a_{2}}$

for all ${\displaystyle a_{1}}$ and ${\displaystyle a_{2}}$ in ${\displaystyle M}$ and all ${\displaystyle g}$ in ${\displaystyle G}$, where ${\displaystyle g\cdot a}$ denotes ${\displaystyle \rho (g,a)}$. A right ${\displaystyle G}$-module is defined similarly. Given a left ${\displaystyle G}$-module ${\displaystyle M}$, it can be turned into a right ${\displaystyle G}$-module by defining ${\displaystyle a\cdot g=g^{-1}\cdot a}$.

A function ${\displaystyle f:M\rightarrow N}$ is called a morphism of ${\displaystyle G}$-modules (or a ${\displaystyle G}$-linear map, or a ${\displaystyle G}$-homomorphism) if ${\displaystyle f}$ is both a group homomorphism and ${\displaystyle G}$-equivariant.

The collection of left (respectively right) ${\displaystyle G}$-modules and their morphisms form an abelian category ${\displaystyle G{\textbf {-Mod}}}$ (resp. ${\displaystyle {\textbf {Mod-}}G}$). The category ${\displaystyle G{\text{-Mod}}}$ (resp. ${\displaystyle {\text{Mod-}}G}$) can be identified with the category of left (resp. right) ${\displaystyle \mathbb {Z} G}$-modules, i.e. with the modules over the group ring ${\displaystyle \mathbb {Z} [G]}$.

A submodule of a ${\displaystyle G}$-module ${\displaystyle M}$ is a subgroup ${\displaystyle A\subseteq M}$ that is stable under the action of ${\displaystyle G}$, i.e. ${\displaystyle g\cdot a\in A}$ for all ${\displaystyle g\in G}$ and ${\displaystyle a\in A}$. Given a submodule ${\displaystyle A}$ of ${\displaystyle M}$, the quotient module ${\displaystyle M/A}$ is the quotient group with action ${\displaystyle g\cdot (m+A)=g\cdot m+A}$.

Examples

• Given a group ${\displaystyle G}$, the abelian group ${\displaystyle \mathbb {Z} }$ is a ${\displaystyle G}$-module with the trivial action ${\displaystyle g\cdot a=a}$.
• Let ${\displaystyle M}$ be the set of binary quadratic forms ${\displaystyle f(x,y)=ax^{2}+2bxy+cy^{2}}$ with ${\displaystyle a,b,c}$ integers, and let ${\displaystyle G={\text{SL}}(2,\mathbb {Z} )}$ (the 2×2 special linear group over ${\displaystyle \mathbb {Z} }$). Define

${\displaystyle (g\cdot f)(x,y)=f((x,y)g^{t})=f\left((x,y)\cdot {\begin{bmatrix}\alpha &\gamma \\\beta &\delta \end{bmatrix}}\right)=f(\alpha x+\beta y,\gamma x+\delta y),}$

where

${\displaystyle g={\begin{bmatrix}\alpha &\beta \\\gamma &\delta \end{bmatrix}}}$

and ${\displaystyle (x,y)g}$ is matrix multiplication. Then ${\displaystyle M}$ is a ${\displaystyle G}$-module studied by Gauss.^[2] Indeed, we have

${\displaystyle g(h(f(x,y)))=gf((x,y)h^{t})=f((x,y)h^{t}g^{t})=f((x,y)(gh)^{t})=(gh)f(x,y).}$

• If ${\displaystyle V}$ is a representation of ${\displaystyle G}$ over a field ${\displaystyle K}$, then ${\displaystyle V}$ is a ${\displaystyle G}$-module (it is an abelian group under addition).

Topological groups

If ${\displaystyle G}$ is a topological group and ${\displaystyle M}$ is an abelian topological group, then a topological G-module is a G-module where the action map ${\displaystyle G\times M\rightarrow M}$ is continuous (where the product topology is taken on ${\displaystyle G\times M}$).^[3]

In other words, a topological G-module is an abelian topological group ${\displaystyle M}$ together with a continuous map ${\displaystyle G\times M\rightarrow M}$ satisfying the usual relations ${\displaystyle g(a+a')=ga+ga'}$, ${\displaystyle (gg')a=g(g'a)}$, and ${\displaystyle 1a=a}$.