G-module

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

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

Definition and basics

Let G {\displaystyle G} {\displaystyle G} be a group. A left G {\displaystyle G} {\displaystyle G}-module consists of[1] an abelian group M {\displaystyle M} {\displaystyle M} together with a left group action ρ : G × M → M {\displaystyle \rho :G\times M\to 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}} {\displaystyle g\cdot (a_{1}+a_{2})=g\cdot a_{1}+g\cdot a_{2}}

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

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

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

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

Examples

  • Given a group G {\displaystyle G} {\displaystyle G}, the abelian group Z {\displaystyle \mathbb {Z} } {\displaystyle \mathbb {Z} } is a G {\displaystyle G} {\displaystyle G}-module with the trivial action g ⋅ a = a {\displaystyle g\cdot a=a} {\displaystyle g\cdot a=a}.
  • Let M {\displaystyle 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}} {\displaystyle f(x,y)=ax^{2}+2bxy+cy^{2}} with a , b , c {\displaystyle a,b,c} {\displaystyle a,b,c} integers, and let G = SL ( 2 , Z ) {\displaystyle G={\text{SL}}(2,\mathbb {Z} )} {\displaystyle G={\text{SL}}(2,\mathbb {Z} )} (the 2×2 special linear group over Z {\displaystyle \mathbb {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),} {\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}}} {\displaystyle g={\begin{bmatrix}\alpha &\beta \\\gamma &\delta \end{bmatrix}}}
and ( x , y ) g {\displaystyle (x,y)g} {\displaystyle (x,y)g} is matrix multiplication. Then M {\displaystyle M} {\displaystyle M} is a G {\displaystyle 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).} {\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} {\displaystyle V} is a representation of G {\displaystyle G} {\displaystyle G} over a field K {\displaystyle K} {\displaystyle K}, then V {\displaystyle V} {\displaystyle V} is a G {\displaystyle G} {\displaystyle G}-module (it is an abelian group under addition).

Topological groups

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

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

Notes

  1. Curtis, Charles W.; Reiner, Irving (1988) [1962]. Representation Theory of Finite Groups and Associative Algebras. John Wiley & Sons. ISBN 978-0-470-18975-7.
  2. Kim, Myung-Hwan (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. 178: 83–93. doi:10.1090/s0002-9947-1973-0338132-7.

References