representations vs modules
Let be a group and a field. Recall that a pair is a representation of over , if is a vector space over and is a linear group action (compare with parent object). On the other hand we have a group algebra , which is a vector space over with as a basis and the multiplication is induced from the multiplication in . Thus we can consider modules over . These two concepts are related.
If is a representation of over , then define a -module by putting as a vector space over and the action of on is given by
It can be easily checked that is indeed a -module.
Analogously if is a -module (with action denoted by ,,”), then the pair is a representation of over , where ,,” is given by
As a simple exercise we leave the following proposition to the reader:
Proposition. Let be a representation of over and let be a -module. Then
This means that modules and representations are the same concept. One can generalize this even further by showing that and are both functors, which are (mutualy invert) isomorphisms of appropriate categories.
|Title||representations vs modules|
|Date of creation||2013-03-22 19:18:59|
|Last modified on||2013-03-22 19:18:59|
|Last modified by||joking (16130)|