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 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).

Let ${\displaystyle G}$be a group. A left ${\displaystyle G}$-module consists of 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 ${\displaystyle g\cdot (m+A)=g\cdot m+A}$.

• 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. 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).

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 ${\displaystyle G\times M}$).

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 ${\displaystyle 1a=a}$.

• Chapter 6 ofWeibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.