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I've heard about this construction on the lecture about higher representation theory:

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.

Is there a good — if possible, non-$sl_2$ — example of such a category $\mathcal A$, minimal categories $V_\lambda$ and braid action which explains why one would have such a construction?

Update: Found the notes of the talk that has two $sl_n$ examples, one from quivers, another from sheaves on the grassmannian, $\mathcal V :=\oplus^n_i D^b\mathop{\rm constr}/\mathop{\rm Gr}(i,n)$.

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".

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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").

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.

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).

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According to the notes, the conjecture doesn't say that any V decomposes as a sum numbered by irreps of g, merely that V decomposes as a sum according to weights and that there's a filtration numbered by irreps. For the reference, the talk in question has notes at math.utexas.edu/users/benzvi/GRASP/lectures/IAS/… –  Ilya Nikokoshev Nov 15 '09 at 0:53
    
Ah, well, this is what happens when you don't explain your notation. The decomposition corresponding to weight spaces is no conjecture; any reasonable version of categorifying Kac-Moody algebras will have that property. The braid action is more interesting though. Rouquier constructs a candidate, but doesn't prove that it satisfies the braid relations. –  Ben Webster Nov 15 '09 at 2:50

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