Modules
With the group algebra in hand, we now define a “
-module” to be a module for the group algebra of . That is, it’s a (finite-dimensional) vector space 
and a bilinear map ![A:\mathbb{C}[G]\times V\to V](https://s0.wp.com/latex.php?latex=A%3A%5Cmathbb%7BC%7D%5BG%5D%5Ctimes+V%5Cto+V&bg=e6e6e6&fg=333333&s=0&c=20201002 "A:\mathbb{C}[G]\times V\to V")
. This map must satisfy 
and 
.
This is really the same thing as a representation, since we may as well pick a basis 
for and write 
. Then for any 
we can write
That is, 
is a linear map from to itself, with its matrix entries given by 
. We define this matrix to be 
, which must be a representation because of the conditions on 
above.
Conversely, if we have a matrix representation 
, we can define a module map for 
as
where we apply the matrix to the column vector 
. This must satisfy the above conditions, since they reflect the fact that 
is a representation.
In fact, to define , all we really need to do is to define it for the basis elements ![\mathbf{g}\in\mathbb{C}[G]](https://s0.wp.com/latex.php?latex=%5Cmathbf%7Bg%7D%5Cin%5Cmathbb%7BC%7D%5BG%5D&bg=e6e6e6&fg=333333&s=0&c=20201002 "\mathbf{g}\in\mathbb{C}[G]")
. Then linearity will take care of the rest of the group algebra. That is, we can just as well say that a -module is a vector space and a function 
satisfying the following three conditions:
- is linear in :
.
- preserves the identity:
.
- preserves the group operation:
.
The difference between the representation viewpoint and the -module viewpoint is that representations emphasize the group elements and their actions, while -modules emphasize the representing space . This viewpoint will be extremely helpful when we want to consider a representation as a thing in and of itself. It’s easier to do this when we think of it as a vector space equipped with the extra structure of a -action.
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September 15, 2010 -
Posted by John Armstrong |
Algebra, Group theory, Representation Theory
For whatever reason, I took a course in Module Theory at Caltech, got lost in the forest, and couldn’t remember by the end what I was learning this FOR. Thanks for being so clear. Maybe I’ll get it better this time around…
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