G-module

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 ''G''. 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 ''R''-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''·(''a''₁ + ''a''₂) = ''g''·''a''₁ + ''g''·''a''₂
and
:(''g''₂ ''x'' ''g''₁)·''a'' = ''g''₂·(''g''₁·''a'')
where ''g''·''a'' denotes ρ(''g'',''a'') and ''x'' denotes the binary operation inside the group ''G''. 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''.

Examples

*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'') = ''ax''² + 2''bxy'' + ''cy''² with ''a'', ''b'', ''c'' integers, and let ''G'' = SL(2, Z) (the 2×2 special linear group over Z). Define
::$(g\cdot f)(x,y)=f((x,y)g^t)=f((x,y)\cdot\begin \alpha & \gamma \\ \beta & \delta \end)=f(\alpha x+\beta y,\gamma x+\delta y),$
:where
::$g=\begin \alpha & \beta \\ \gamma & \delta \end$
: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''×''M'' → ''M'' is continuous (where the product topology is taken on ''G''×''M'').

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 1''a'' = ''a''.

Notes

References

*Chapter 6 of {{Weibel IHA

Group theory
Representation theory of groups