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).

Definition and basics

Let $G$ be a group. A left $G$-module consists of an abelian group $M$ together with a left group action $\rho:G\times M\to M$ such that
:$g\cdot(a_1+a_2)=g\cdot a_1+g\cdot a_2$
for all $a_1$ and $a_2$ in $M$ and all $g$ in $G$, where $g\cdot a$ denotes $\rho(g,a)$. A right $G$-module is defined similarly. Given a left $G$-module $M$, it can be turned into a right $G$-module by defining $a\cdot g=g^\cdot a$.

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

The collection of left (respectively right) $G$-modules and their morphisms form an abelian category $G\textbf$ (resp. $\textbfG$). The category $G\text$ (resp. $\textG$) can be identified with the category of left (resp. right) $\mathbbG$-modules, i.e. with the modules over the group ring \mathbb /math>.

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

Examples

*Given a group $G$, the abelian group $\mathbb$ is a $G$-module with the ''trivial action'' $g\cdot a=a$.
*Let $M$ be the set of binary quadratic forms $f(x,y)=ax^2+2bxy+cy^2$ with $a,b,c$ integers, and let $G=\text(2,\mathbb)$ (the 2×2 special linear group over $\mathbb$). Define
::$(g\cdot f)(x,y)=f((x,y)g^t)=f\left((x,y)\cdot\begin \alpha & \gamma \\ \beta & \delta \end\right)=f(\alpha x+\beta y,\gamma x+\delta y),$
:where
::$g=\begin \alpha & \beta \\ \gamma & \delta \end$
:and $(x,y)g$ is matrix multiplication. Then $M$ is a $G$-module studied by Gauss. Indeed, we have

:: $g(h(f(x,y))) = gf((x,y)h^t)=f((x,y)h^tg^t)=f((x,y)(gh)^t)=(gh)f(x,y).$

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

Topological groups

If $G$ is a topological group and $M$ is an abelian topological group, then a topological ''G''-module is a ''G''-module where the action map $G\times M\rightarrow M$ is continuous (where the product topology is taken on $G\times M$).

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

Notes

References

*Chapter 6 of {{Weibel IHA

Group theory
Representation theory of groups