When are modules and representations not the same thing?

Question

I've been trying for a while to get a real concrete handle on the relationship between representations and modules. To frame the question, I'll put here the standard situation I have in mind:

A ring $R$ lives in the category Ab of Abelian groups as an internal monoid $(\mu_R, \eta_R)$. A module is then just an Abelian group $A$ and a map $m : R \otimes A \rightarrow A$ that commutes with the monoid structure in the way you'd expect.

Alternatively, take an Abelian group $A$ and look at its group of endomorphisms $[A,A]$. This has an internal monoid $(\mu_A, \eta_A)$ just taking composition and identity. Then a representation is just a monoid homomorphism $(R, \mu_R, \eta_R) \rightarrow (A, \mu_A, \eta_A)$ in Ab. I.e. a ring homomorphism.

But then, Ab is monoidal closed, so these are the same concept under the iso

$$\hom(R\otimes A, A) \cong \hom(R, [A,A])$$

This idea seems to work for any closed category where one wants to relate a multiplication to composition. So, my question is, since these things are isomorphic in such a general context, why are they taught as two separate concepts? Is it merely pedagogical, or are there useful examples where modules and representations are distinct?

Answer 1

Here is my representation theorist's perspective: the key difference between representations and modules is that representations are "non-linear", whereas modules are "linear". I'll concentrate on the case of groups as the most familiar, but this applies more generally.

As Greg has already mentioned, in the most general sense, a representation is a homomorphism $f:G\to H,$ and usually there is no linear (or additive) structure on $H$, i.e. the set $f(g)$ need not be closed under sums; in fact, if $H$ is a non-abelian group, e.g. the symmetric group, the notion of sum doesn't even make sense (if $H=GL(V)$ then we may view its elements as endomorphisms of $V$ and add them, but this is unnatural since, by definition, $f$ is compatible with multiplicative structure). By contrast, a module involves a linear action $G\times V\to V,$ which is then "completed" by allowing arbitrary linear combinations, leading to certain technical advantages.

Here is an example of a construction that is very useful and makes perfect sense module-theoretically, but not representation-theoretically: change of scalars. Given a module $M$ over a group ring $R[G]$ and a commutative ring homomorphism $R\to S,$ one gets a module $S\otimes_R M$ over the group ring $S[G]$. Common examples involve extensions of scalars (e.g. from $\mathbb{R}$ to $\mathbb{C}$, from a field $K$ of definition to the splitting field, from $\mathbb{Z}$ to $\mathbb{Z}_p$) and, more to the point, reductions (e.g. from $\mathbb{Z}$ or $\mathbb{Z}_p$ to $\mathbb{Z}/p\mathbb{Z}$). The module language is, predictably, also very useful in providing categorical descriptions of various operations on representations, such as functors of induction and restriction,

$$Ind_H^G: H\text{-mod}\to G\text{-mod}\ \text{ and }\ Res_H^G: G\text{-mod}\to H\text{-mod},$$

where $H$ is a subgroup of $G,$ or the monoidal structure on $G$-mod.

Finally, here are two illustrations of the complementary nature of the two approaches besides the group case, in linear algebra. A single linear transformation $T:V\to V$ on a finite-dimensional vector space $V$ over $K$ is most naturally viewed as a representation (no additive structure); in this case, it's a representation of the quiver with a single vertex and a single loop. From this point of view, classification up to isomorphism is a problem about conjugacy classes of linear transformations,

$$T\to gTg^{-1},\ g\in GL(V).$$

By contrast, in the module style description we associate with $T$ a module over the ring $K[x]$ of polynomials in one variable over $K$ and classification problem reduces to the structure of modules over $K[x]$, which is a PID, with all the usual consequences. (Here the module picture is more illuminating.) If we consider a linear operator $S:V\to W$ between two different vector spaces,

$$S\to hSg^{-1},\ g\in GL(V),\ h\in GL(W),$$

and a classification up to isomorphism is accomplished by row and column reduction. The corresponding quiver $\circ\to\circ$ is a single arrow connecting two distinct vertices, but its path algebra is less familiar. (Here the representation theory picture is more illuminating.)

Answer 2

To expand on Tom's answer, the word "representation" is a 19th century word that originally meant "group homomorphism". If $f:G \to H$ is a homomorphism from a group $G$ to a group $H$, then $f(g)$ "represents" the element $g$. $H$ is taken to be a "familiar" or "explicit" group, usually a matrix group but also sometimes a permutation group.

The word "module" is a 20th century word (I think) that means "generalized vector space".

As has been pointed out, a representation of a group $G$ is equivalent to a $k[G]$-module. These days the terms are largely interchangeable; you can also talk about a representation of an algebra instead of a group. Certainly you can add topology to the conditions, for instance by using the group $C^*$-algebra of a locally compact topological group.

To the extent that there is still a useful distinction, there is a difference in emphasis. If a ring $R$ (or a group or whatever) acts on an abelian group $A$, and you consider its action to be a low-level structure, analogous to multiplying a vector by a scalar, then you should call $A$ an $R$-module. On the other hand, if you think of the action as a high-level geometric effect, analogous to a group acting on a manifold, then you should call it a representation. If you don't care, then you can use either term or both and it's all cool AFAIK :-). Possibly the word "module" is slowly supplanting the word "representation", because it's a shorter word as well as more modern and more general.

Answer 3

It is certainly true that the category of representations of a group $G$ over a field $k$ is equivalent to the category of modules for the group ring $k[G]$, and it is often productive to rephrase questions about representations about questions about modules. Below, I give some examples of structure which is easier to discuss in terms of representations. But, as I will indicate, it is usually possible to rephrase in terms of modules with enough effort.

Tensor products: If $V$ and $W$ are two representations of $G$, then $V \otimes W$ has a natural structure as a $G$-representation. For $k[G]$ modules, this is not true; the tensor product has to be added as additional structure on the category $k[G]$-rep. Here is an explicit example: Let $G=\mathbb{Z}/4$ and let $H = \mathbb{Z}/2 \times \mathbb{Z}/2$. Then $\mathbb{C}[G]$ and $\mathbb{C}[H]$ are isomrphic rings, but the tensor structures on $\mathbb{C}[G]$-modules and $\mathbb{C}[H]$-modules are inequivalent. The same issue exists with duals. People who like rings better than groups would say that the issue is that I am talking about the algebra structure of $k[G]$ when I should be talking about the Hopf algebra structure.

Topology: Suppose that $G$ is a topological group (maybe a Lie group) and $k$ a toplogical field (maybe $\mathbb{R}$). Then a continuous representation of $G$ is a map $G \times V \to V$ which is a group action, $k$-linear, and continuous. I imagine there is a way to put a topology on $k[G]$ so that a continuous representation is a $k[G]$-module such that $k[G] \times V \to V$ is continuous, but I haven't seen it. And this will get worse with adjectives like smooth, algebraic, ...

Answer 4

I would teach that an $R$-module is an abelian group $A$ plus a map of sets $R\times A\to A$ satisfying certain identities. I would probably also point out that this is the same thing as an abelian group $A$ plus a ring homomorphism $R\to End(A)$. I would also point out that "vector space" is the traditional term for "module" when $R$ is a field.

Similarly, I would teach that an action of a group $G$ on a set $X$ is a map of sets $G\times X\to X$ satisfying certain identities, and I would probably also point out that that is the same as a group homomorphism $G\to Aut(X)$; and if it seemed appropriate for those students I would also say that this second point of view is useful for generalizing the idea so as to make groups act on things other than sets.

An action of a group $G$ on a $k$-module is the same as a module for the group ring $kG$. You can also call this a representation of $G$ over $k$. This is not traditionally called a representation of $kG$.

The fact that there are overlapping definitions is just historical accident. The word "module" was in use in special cases long before there was category theory, even before there was abstract ring theory as we know it. So was the word "representation". The fact that the two terms are both still used is not because somebody decided on a good reason to keep them both; they just survived, as words do.