G-module

In mathematics, given a group G {\displaystyle G} , a G-module is an abelian group M {\displaystyle M} on which G {\displaystyle G} acts compatibly with the abelian group structure on 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 G {\displaystyle G} -modules.

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

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

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

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

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

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

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

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

( g ⋅ f ) ( x , y ) = f ( ( x , y ) g t ) = f ( ( x , y ) ⋅ [ α γ β δ ] ) = f ( α x + β y , γ x + δ 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

g = [ α β γ δ ] {\displaystyle g={\begin{bmatrix}\alpha &\beta \\\gamma &\delta \end{bmatrix}}}

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

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

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

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