## Categorifying the group representations

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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".

-

## 1 Answer

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).

-

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 2009 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 2009 at 2:50
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 15, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9385179877281189, "perplexity_flag": "middle"}