G-module

Let G be a group. A left G-module consists of[1] an abelian group M together with a left group action ρ: G × M → M such that

g·(a + b) = g·a + g·b

where g·a denotes ρ(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·g = g⁻¹·a.

A function f : M → 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-Mod (resp. Mod-G). The category G-Mod (resp. Mod-G) can be identified with the category of left (resp. right) ZG-modules, i.e. with the modules over the group ring Z[G].

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

• Given a group G, the abelian group Z is a G-module with the trivial action g·a = a.
• Let M be the set of binary quadratic forms f(x, y) = ax² + 2bxy + cy² with a, b, c integers, and let G = SL(2, Z) (the 2×2 special linear group over Z). Define

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

where

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

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

${\displaystyle g(h(f(x,y))=gf((x,y)h^{t})=f((x,y)h^{t}g^{t})=f((x,y)(gh)^{t})=(gh)f(x,y).}$

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

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×M → M is continuous (where the product topology is taken on G×M).[3]

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

• Chapter 6 of Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.