Categorifying the group representations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T00:43:35Z http://mathoverflow.net/feeds/question/5568 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5568/categorifying-the-group-representations Categorifying the group representations Ilya Nikokoshev 2009-11-14T20:19:14Z 2009-11-15T01:07:38Z <p>I've heard about this construction on the lecture about <strong>higher representation theory</strong>:</p> <blockquote> <p>Given a Lie algebra $g$, one constructs $\mathcal A$, a category whose $K_0$ is the universal enveloping algebra of $g$. Conjecture: any $\mathcal A$-acted triangulated category $\mathcal V$ (with its $K$ locally finite) decomposes to $\oplus \mathcal V_\lambda$ with braid action; and there is bijection between $g$-representations and minimal such categories.</p> </blockquote> <p>Is there a good &mdash; if possible, non-$sl_2$ &mdash; example of such a category $\mathcal A$, minimal categories $V_\lambda$ and braid action which explains why one would have such a construction?</p> <p><strong>Update:</strong> Found the <a href="http://www.math.utexas.edu/users/benzvi/GRASP/lectures/IAS/rouquierhigher.pdf" rel="nofollow">notes of the talk</a> that has two $sl_n$ examples, one from quivers, another from sheaves on the grassmannian, $\mathcal V :=\oplus^n_i D^b\mathop{\rm constr}/\mathop{\rm Gr}(i,n)$. </p> <p>A more accessible text for either example would be welcome! Because if the best way to understand these is to "get" quantum groups, that's quite a big topic. My idea was more like "maybe this is a good place to start". </p> http://mathoverflow.net/questions/5568/categorifying-the-group-representations/5592#5592 Answer by Ben Webster for Categorifying the group representations Ben Webster 2009-11-14T23:45:10Z 2009-11-14T23:45:10Z <p>The $sl_n$ version of this shouldn't so bad. I think it's just self-dual objects in parabolic category O and shuffling functors, though I'll admit, I haven't checked this myself, and doubt it's written properly somewhere. Probably the best reference is the papers of Brundan and Kleshchev (for example "Schur-Weyl duality for higher levels"). </p> <p>I suspect the inspiration for such a conjecture isn't really particular examples so much a philosophy about what sort of structures on a quantum group should be categorifiable.</p> <p>By the way, I think your conjecture might be a bit too strong (at least as I interpret it). The 2-representations of a 2-Kac-Moody algebra aren't semi-simple (I've got a huge supply of non-semisimple examples categorifying tensor products).</p>