G-module

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). 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 . 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 -module on which G acts linearly (i.e. as a group of R -module automorphisms).

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).]] 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 . 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 -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^{-1}\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{-Mod} (resp. \textbf{Mod-}G ). The category G\text{-Mod} (resp. \text{Mod-}G ) can be identified with the category of left (resp. right) \mathbb{Z}G -modules, i.e. with the modules over the group ring \mathbb{Z}[G] . 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{Z} 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{SL}(2,\mathbb{Z}) (the 2×2 special linear group over \mathbb{Z} ). Define (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=\begin{bmatrix} \alpha & \beta \\ \gamma & \delta \end{bmatrix} 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

Text derived from https://en.wikipedia.org/w/index.php?oldid=1320979952 under CC BY-SA 4.0; modifications have been made.