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?