Indecomposable $\mathfrak{sl}_3$-representations

Question

Motivation: Consider $\mathfrak{sl}_2$. Given an integral weight $\lambda\in\mathbf Z$, then the only indecomposable modules contained in $\mathcal O_\lambda$ are the Verma modules and the simple modules with highest weight linked to $\lambda$ as well es their projective covers and injective hulls, cf. for example prop. 3.12 in Humphreys’ category .

Question: Do similar statements hold more generally? At least for $\mathfrak{sl}_3$ or $\mathfrak{sl}_n$? I doubt, but cannot find any clear statements.

Does the paper below answer your question? – Hanno Apr 18 '17 at 05:54 — Hanno, Apr 18 '17 at 05:54

score 1 · Accepted Answer · answered Apr 14 '17 at 14:13

1

The representation types of the blocks of ${\mathscr O}$ have been completely determined here.

answered Apr 14 '17 at 14:13

Hanno

19,510

I'm not sure how to correctly understand the paper. According to table 2, $\mathcal O_\lambda$ for ₃ has finite representation type (does this mean there are finitely many indecomposable modules?) iff ₀ = A₁, which is the case if lies on a reflection plane. For regular , no such list exists. Is this the correct meaning of that paper? – Bubaya Apr 19 '17 at 17:55

Indecomposable $\mathfrak{sl}_3$-representations

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