G-module

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

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle g(h(f(x,y)))=gf((x,y)h^{t})=f((x,y)h^{t}g^{t})=f((x,y)(gh)^{t})=(gh)f(x,y).}${\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 ${\displaystyle V}${\displaystyle V} is a representation of ${\displaystyle G}${\displaystyle G} over a field ${\displaystyle K}${\displaystyle K}, then ${\displaystyle V}${\displaystyle V} is a ${\displaystyle G}${\displaystyle G}-module (it is an abelian group under addition).

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

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

• Chapter 6 of Weibel, 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.

• Group theory
• Representation theory of groups

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