Timeline for How complicated is infinite-dimensional "undergraduate linear algebra"?
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| Oct 22, 2010 at 3:57 | comment | added | Theo Johnson-Freyd | @Jim: Well, that's some data about how widely the term "undergraduate linear algebra" has spread as code for "representation theory of K[x]". I first heard the line in a talk about quivers: in general, the representation theory of a quiver is completely intractable; for simply-laced quivers it's doable, and for the quiver with one object and one arrow (from the object to itself) you get precisely this category. | |
| Oct 21, 2010 at 22:37 | comment | added | Jim Humphreys | The question itself has real content, but I am suitably amused by the descriptions of "high school" and "undergraduate" linear algebra. Leaving aside my own inadequate education at both levels, today's high school and college students rarely encounter linear algebra in that spirit. At UMass the engineering students (and others stuck in the same classes) learn mainly a few unrealistic algorithms for 3x3 matrices without any conceptual foundation. And how many U.S. high school students even hear the magic words "vector space"? College students just ask whether it's on the exam. (No.) | |
| Oct 18, 2010 at 18:12 | comment | added | Theo Johnson-Freyd | @BCnrd: There are precisely two linear symmetric monoidal continuous cocontinuous functors from finite-dimensional-Rep(N) to Vect (both over C), the "forgetful" one and the one that throws away the 0 generalized eigenspace. I wanted to know if this continues to the infinite-dimensional representation theory. | |
| Oct 18, 2010 at 18:11 | comment | added | Theo Johnson-Freyd | @Emerton: "any C[x]-module is the direct limit of its f.g. submodules" almost completely answers the question, and does answer my post-question paragraph. As a description of the full category, one would need a list of all singly-generated modules (easy, as C[x] is a PID) and some description of the extension problem (complicated, per Greg Kuperberg's answer). | |
| Oct 18, 2010 at 13:37 | answer | added | Greg Kuperberg | timeline score: 12 | |
| Oct 18, 2010 at 10:01 | comment | added | KConrad | Hailong, there are books by Kaplansky and by Fuchs with the same relevant title: "Infinite Abelian Groups". I've not looked at either one, though. | |
| Oct 18, 2010 at 6:40 | comment | added | Hailong Dao | Dear BCnrd, do you have some reference for the "finite rank Z-modules" stuff? I am really curious, thanks. | |
| Oct 18, 2010 at 4:10 | comment | added | BCnrd | Andy, perhaps Harry has in mind that $F[x]$-modules are the quasi-coherent sheaves on the affine $F$-line, and he conflates "geometry" with "sheaf theory"? Anyway, this question is long on setup and short on motivation. There is (or used to be?) a school of mathematicians who specialized in the study of "finite rank $\mathbf{Z}$-modules", meaning torsion-free abelian subgroups of finite-dimensional $\mathbf{Q}$-vector spaces. A "general" such object doesn't have a particularly explicit description, and I imagine for $F[x]$ it is similar. Emerton's comment hits the nail on the head, as usual | |
| Oct 18, 2010 at 3:50 | comment | added | Emerton | ... submodules of the form $\mathbb C[x]$, are exactly shift operators, which don't admit eigenvectors, and thinking of them from this spectral-theoretic point of view is often helpful.) | |
| Oct 18, 2010 at 3:49 | comment | added | Emerton | I think I'm a bit confused by this question. Given your last paragraph, why isn't an answer given by saying that any $\mathbb C[x]$-module is the direct limit of its f.g. submodules, and then observe that these are classified. This answer is not meant to be facetious, by the way: in my research I frequently have to deal with problems involving the structure of matrices acting on infinite-dimensional vector spaces, in contexts that are hybrid between analysis and pure algebra, and this way of thinking is basic to how I proceed in these contexts. (For example, free factors, i.e. ... | |
| Oct 18, 2010 at 3:41 | comment | added | Andy Putman | @Harry : Huh??? | |
| Oct 18, 2010 at 3:39 | comment | added | Harry Gindi | Doesn't that basically translate into studying the geometry of the affine line? | |
| Oct 18, 2010 at 3:31 | history | asked | Theo Johnson-Freyd | CC BY-SA 2.5 |