G-module

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

\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

References

1. (1988). "Representation Theory of Finite Groups and Associative Algebras". John Wiley & Sons.
2. (1999). "Integral Quadratic Forms and Lattices: Proceedings of the International Conference on Integral Quadratic Forms and Lattices, June 15–19, 1998, Seoul National University, Korea". American Mathematical Soc..
3. D. Wigner. (1973). "Algebraic cohomology of topological groups". Trans. Amer. Math. Soc..