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). Toroidal coord.png
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, 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.

Contents

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 be a group. A left -module consists of [1] an abelian group together with a left group action such that

for all and in and all in , where denotes . A right -module is defined similarly. Given a left -module , it can be turned into a right -module by defining .

A function is called a morphism of -modules (or a -linear map, or a -homomorphism) if is both a group homomorphism and -equivariant.

The collection of left (respectively right) -modules and their morphisms form an abelian category (resp. ). The category (resp. ) can be identified with the category of left (resp. right) -modules, i.e. with the modules over the group ring .

A submodule of a -module is a subgroup that is stable under the action of , i.e. for all and . Given a submodule of , the quotient module is the quotient group with action .

Examples

where
and is matrix multiplication. Then is a -module studied by Gauss. [2] Indeed, we have

Topological groups

If is a topological group and is an abelian topological group, then a topological G-module is a G-module where the action map is continuous (where the product topology is taken on ). [3]

In other words, a topological G-module is an abelian topological group together with a continuous map satisfying the usual relations , , and .

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