Timeline for When are modules and representations not the same thing?
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| Jul 20, 2010 at 17:30 | comment | added | Victor Protsak | Note: positive-dimensional vector spaces over non-discrete locally compact fields are never compact. There are some genuine difficulties in coming up with the correct notion of a group algebra in the topological setting that will faithfully capture representation-theoretic picture (see Kirillov's "Elements of representation theory"). Many of them concern a good category to which it should belong. The reduced $C^*$-algebra of an infinite discrete group is a good illustration. This cannot be fully computed even in easy cases. | |
| Jul 20, 2010 at 15:48 | comment | added | Aleks Kissinger | TOPOLOGY: Representations of top. groups is probably a good foil for the mod/rep correspondence. For one thing Top fails to be cartesian closed, so the first crack at comparing these things by chasing the iso hom(GxV,V) \cong hom(G,[V,V]) fails unless V and G are nice spaces (compact Hausdorff or something). TENSORS: It seems the natural representation of G on V (x) W already (implicitly) uses the comultiplication from the Hopf algebra structure of k[G], i.e. the induced representation is "(psi (x) phi) o delta" ...where delta is the linear map that copies the basis elements of k[G]. | |
| Jul 20, 2010 at 15:48 | comment | added | George McNinch | Concerning your last sentence ('this will get worse with adjectives like ... algebraic'), "algebraic" representations of an affine group scheme $G$ over a field $k$ are actually co-modules for the Hopf algebra of regular functions $k[G]$. This means that the action of $G$ on the representation space $V$ is defined by a mapping $V \to k[G] \otimes V$ satisfying some natural diagrams. When $G$ is smooth and $k$ is alg. closed, one can of course view $V$ as a module for the group algebra of the "abstract" group $G(k)$, but as you point out, it isn't clear this is useful. | |
| Jul 20, 2010 at 14:34 | history | edited | David E Speyer | CC BY-SA 2.5 |
added 30 characters in body
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| Jul 20, 2010 at 14:15 | history | answered | David E Speyer | CC BY-SA 2.5 |