G-module

Let $G$ be a group. A left $G$-module consists of{{cite book |title=Representation Theory of Finite Groups and Associative Algebras |year=1988 |first1=Charles W. |last1=Curtis |first2=Irving |last2=Reiner |author1-link=Charles W. Curtis |author2-link=Irving Reiner |publisher=John Wiley & Sons |orig-year=1962 |isbn=978-0-470-18975-7 |url-access=registration |url=https://archive.org/details/representationth11curt}} an abelian group $M$ together with a left group action $\backslash rho:G\backslash times\; M\backslash to\; M$ such that

:$g\backslash cdot(a\_1+a\_2)=g\backslash cdot\; a\_1+g\backslash cdot\; a\_2$

for all $a\_1$ and $a\_2$ in $M$ and all $g$ in $G$, where $g\backslash cdot\; a$ denotes $\backslash rho(g,a)$. A right $G$-module is defined similarly. Given a left $G$-module $M$, it can be turned into a right $G$-module by defining $a\backslash cdot\; g=g^\{-1\}\backslash cdot\; a$.

A function $f:M\backslash rightarrow\; N$ 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 $G\backslash textbf\{-Mod\}$ (resp. $\backslash textbf\{Mod-\}G$). The category $G\backslash text\{-Mod\}$ (resp. $\backslash text\{Mod-\}G$) can be identified with the category of left (resp. right) $\backslash mathbb\{Z\}G$-modules, i.e. with the modules over the group ring $\backslash mathbb\{Z\}[G]$.

A submodule of a $G$-module $M$ is a subgroup $A\backslash subseteq\; M$ that is stable under the action of $G$, i.e. $g\backslash cdot\; a\backslash in\; A$ for all $g\backslash in\; G$ and $a\backslash in\; A$. Given a submodule $A$ of $M$, the quotient module $M/A$ is the quotient group with action $g\backslash cdot\; (m+A)=g\backslash cdot\; m+A$.

• Given a group $G$, the abelian group $\backslash mathbb\{Z\}$ is a $G$-module with the trivial action $g\backslash cdot\; a=a$.
• Let $M$ be the set of binary quadratic forms $f(x,y)=ax^2+2bxy+cy^2$ with $a,b,c$ integers, and let $G=\backslash text\{SL\}(2,\backslash mathbb\{Z\})$ (the 2×2 special linear group over $\backslash mathbb\{Z\}$). Define

::$(g\backslash cdot\; f)(x,y)=f((x,y)g^t)=f\backslash left((x,y)\backslash cdot\backslash begin\{bmatrix\}$

\alpha & \gamma \\

\beta & \delta

\end{bmatrix}\right)=f(\alpha x+\beta y,\gamma x+\delta y),

:where

::$g=\backslash begin\{bmatrix\}$

\alpha & \beta \\

\gamma & \delta

\end{bmatrix}

:and $(x,y)g$ is matrix multiplication. Then $M$ is a $G$-module studied by Gauss.{{Citation| title= Integral Quadratic Forms and Lattices: Proceedings of the International Conference on Integral Quadratic Forms and Lattices, June 15–19, 1998, Seoul National University, Korea|first1=Myung-Hwan |last1= Kim |publisher=American Mathematical Soc.|year=1999}} 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).

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 $G\backslash times\; M\backslash rightarrow\; M$ is continuous (where the product topology is taken on $G\backslash times\; M$).{{cite journal| title= Algebraic cohomology of topological groups| author=D. Wigner| journal= Trans. Amer. Math. Soc.| volume=178 |year=1973|pages=83–93| doi=10.1090/s0002-9947-1973-0338132-7| doi-access=free}}

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

{{Reflist}}

• Chapter 6 of {{Weibel IHA}}

Category:Group theory

Category:Representation theory of groups