Timeline for ubiquity, importance of path algebras
Current License: CC BY-SA 2.5
9 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Nov 15, 2009 at 0:34 | comment | added | Dave Penneys | @Mariano and @Yemon - Yes, we are talking about two different types of path algebras. The ones I'm talking about are due to Ocneanu and Sunder, and they are useful in analyzing inclusions of multi-matrix algebras. See the above references in my response to @Yemon for more details. @Mariano - My answer is probably not the best answer to this question. I gave an answer to why path algebras are useful to subfactors, planar algebras, and $C^\ast$-algebras. | |
| Nov 14, 2009 at 23:03 | comment | added | Mariano Suárez-Álvarez | I really cannot see how this is an answer to the question. It is rather rare that a path algebra be semisimple: as soon as there is an arrow in the quiver, $Ext^1$ is non-zero... Maybe there are two different meanings of "path algebra" being mixed, but in the context of Gabriel's theorem, a path algebra is isomorphic to a semisimple algebra iff there are no arrows in the quiver. | |
| Nov 14, 2009 at 22:54 | comment | added | Yemon Choi | @Dave: I think I am misunderstanding what "path algebra" means, so I need to go and look it up. But you are right (and I was overly glib) in that tracking the inclusions of subalgebras is interesting (cf. Bratteli diagrams) | |
| Nov 14, 2009 at 21:59 | comment | added | Dave Penneys | @Yemon - they are very interesting indeed. If you are interested in inclusions of finite von Neumann algebras, they are instrumental in calculating the basic construction, conditional expectations, Pimsner-Popa bases, etc. See Jones and Sunder's book Introduction to Subfactors or Goodman, de la Hapre, and Jones' book Coxeter Graphs and Towers of Algebras for all sorts of subfactor applications, including commuting squares, Ocneanu compactness, and more. | |
| Nov 14, 2009 at 21:51 | comment | added | Emily Peters | But I'm generally suspicious of bases ... | |
| Nov 14, 2009 at 21:50 | vote | accept | Emily Peters | ||
| Nov 15, 2009 at 17:31 | |||||
| Nov 14, 2009 at 20:55 | comment | added | Yemon Choi | I'm not sure direct sums of matrix algebras (which, by Wedderburn, is what you're talking about in your answer) are particularly interesting examples of path algebras... | |
| Nov 14, 2009 at 19:01 | history | answered | Dave Penneys | CC BY-SA 2.5 |