Basic algebra of $\mathcal{O}_0(\mathfrak{sl}_n(\mathbb{C}))$ — Reference request

Question

It is well known that the principal block $\mathcal{O}_0$ of the BGG category $\mathcal{O}$ of a semisimple Lie algebra is equivalent to the category of finitely generated modules over a certain Koszul algebra. Namely, if we let $W$ denote the Weil group and take $P=\bigoplus_{w\in W}P_w$ to be a projective generator in $\mathcal{O}_0$ (Where $P_w$ is the projective cover of the simple module $L_w:=L(w\cdot 0)$), then denoting $A:=\operatorname{End}_{\mathcal{O}_0}(P)^{\text{op}}$ we get that $A$ is a Koszul algebra and the functor $\operatorname{Hom}(P,*):\mathcal{O}_0\to A\operatorname{-mod}^{\text{fg}}$ is an equivalence.

I was wondering if there are any more explicit descriptions of this algebra (In terms of some of more "classical" algebras, or even just in terms of generators and relations, etc.). All of these things are very classical so there probably are such descriptions out there but I didn't manage to find any.

Any references would be greatly appreciated!

Answer 1

You can find quiver and relations (not sure if they are admissible always) here: http://www.math.uni-bonn.de/ag/stroppel/Quivers.pdf In particular the explicit algebra is only fully understoof for $sl_4$ and below. See also https://link.springer.com/article/10.1007/s00222-006-0005-2 which gives results on the calculation on the quiver and relations for those blocks. It is an open question whether the field coefficients can always be choosen to be +1 or -1, see the end of this article.