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
. 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
. 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.
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…
[…] Actions and Representations From the module perspective, we’re led back to the concept of a group action. This is like a -module, but […]
Pingback by Group Actions and Representations « The Unapologetic Mathematician | September 16, 2010 |
[…] course, this shouldn’t really surprise us. After all, representations of are equivalent to modules for the group algebra; and the very fact that is an algebra means that it comes with a bilinear […]
Pingback by The (Left) Regular Representation « The Unapologetic Mathematician | September 17, 2010 |
[…] Between Representations Since every representation of is a -module, we have an obvious notion of a morphism between them. But let’s be explicit about […]
Pingback by Morphisms Between Representations « The Unapologetic Mathematician | September 21, 2010 |
[…] We say that a module is “reducible” if it contains a nontrivial submodule. Thus our examples last time show […]
Pingback by Reducibility « The Unapologetic Mathematician | September 23, 2010 |
[…] I’d like to cover a stronger condition than reducibility: decomposability. We say that a module is “decomposable” if we can write it as the direct sum of two nontrivial submodules […]
Pingback by Decomposability « The Unapologetic Mathematician | September 24, 2010 |
[…] and Kernels A nice quick one today. Let’s take two -modules and . We’ll write for the vector space of intertwinors from to . This is pretty […]
Pingback by Images and Kernels « The Unapologetic Mathematician | September 29, 2010 |
[…] Now that we know that images and kernels of -morphisms between -modules are -modules as well, we can bring in a very general […]
Pingback by Schur’s Lemma « The Unapologetic Mathematician | September 30, 2010 |
[…] and Commutant Algebras We will find it useful in our study of -modules to study not only the morphisms between them, but the structures that they […]
Pingback by Endomorphism and Commutant Algebras « The Unapologetic Mathematician | October 1, 2010 |
[…] way of looking at it: remember that a representation of a group on a space can be regarded as a module for the group algebra . If we then add a commuting representation of a group , we can actually […]
Pingback by Representing Product Groups « The Unapologetic Mathematician | November 1, 2010 |
[…] that we’re interested in concrete actions of Lie algebras on vector spaces, like we were for groups. Given a Lie algebra we define an -module to be a vector space equipped with a bilinear function […]
Pingback by Lie Algebra Modules « The Unapologetic Mathematician | September 12, 2012 |