representations vs modules

Let G be a group and k a field. Recall that a pair (V,) is a representation of G over k, if V is a vector space over k and :G×VV is a linear group action (compare with parent object). On the other hand we have a group algebraMathworldPlanetmathPlanetmath kG, which is a vector space over k with G as a basis and the multiplication is induced from the multiplication in G. Thus we can consider modules over kG. These two conceptsMathworldPlanetmath are related.

If 𝕍=(V,) is a representation of G over k, then define a kG-module 𝕍¯ by putting 𝕍¯=V as a vector space over k and the action of kG on 𝕍¯ is given by


It can be easily checked that 𝕍¯ is indeed a kG-module.

Analogously if M is a kG-module (with action denoted by ,,”), then the pair M¯=(M,) is a representation of G over k, where ,,” is given by


As a simple exercise we leave the following propositionPlanetmathPlanetmath to the reader:

Proposition. Let 𝕍 be a representation of G over k and let M be a kG-module. Then


This means that modules and representations are the same concept. One can generalize this even further by showing that ¯ and ¯ are both functorsMathworldPlanetmath, which are (mutualy invert) isomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of appropriate categoriesMathworldPlanetmath.

Therefore we can easily define such concepts as ,,direct sum of representations” or ,,tensor productPlanetmathPlanetmath of representations”, etc.

Title representations vs modules
Canonical name RepresentationsVsModules
Date of creation 2013-03-22 19:18:59
Last modified on 2013-03-22 19:18:59
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Definition
Classification msc 20C99