The Unapologetic Mathematician

Mathematics for the interested outsider

Modules

With the group algebra in hand, we now define a “G-module” to be a module for the group algebra of G. That is, it’s a (finite-dimensional) vector space V and a bilinear map A:\mathbb{C}[G]\times V\to V. This map must satisfy A(\mathbf{e},v)=v and A(\mathbf{g},A(\mathbf{h},v))=A(\mathbf{gh},v).

This is really the same thing as a representation, since we may as well pick a basis \{e_i\} for V and write V=\mathbb{C}^d. Then for any g\in G we can write

\displaystyle A(\mathbf{g},e_i)=\sum\limits_{j=1}^dm_i^je_j

That is, A(\mathbf{g},\underbar{\hphantom{X}}) is a linear map from V to itself, with its matrix entries given by m_i^j. We define this matrix to be \rho(g), which must be a representation because of the conditions on A above.

Conversely, if we have a matrix representation \rho:G\to GL_d, we can define a module map for \mathbb{C}^d as

\displaystyle A(\mathbf{g},v)=\rho(g)v

where we apply the matrix \rho(g) to the column vector v. This must satisfy the above conditions, since they reflect the fact that \rho is a representation.

In fact, to define A, all we really need to do is to define it for the basis elements \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 G-module is a vector space V and a function A:G\times V\to V satisfying the following three conditions:

  • A is linear in V: A(g,cv+dw)=cA(g,v)+dA(g,w).
  • A preserves the identity: A(e,v)=v.
  • A preserves the group operation: A(g,A(h,v))=A(gh,v).

The difference between the representation viewpoint and the G-module viewpoint is that representations emphasize the group elements and their actions, while G-modules emphasize the representing space V. 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 G-action.

September 15, 2010 - Posted by | Algebra, Group theory, Representation Theory

11 Comments »

  1. 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…

    Comment by Jonathan Vos Post | September 15, 2010 | Reply

  2. […] 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 | Reply

  3. […] 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 | Reply

  4. […] 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 | Reply

  5. […] 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 | Reply

  6. […] 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 | Reply

  7. […] 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 | Reply

  8. […] 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 | Reply

  9. […] 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 | Reply

  10. […] 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 | Reply

  11. […] 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 | Reply


Leave a comment

« Previous | Next »