Reference request for indecomposable representations of $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$

Question

Is there a good reference on the classification of indecomposable representations of the Lie algebra $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$ (of course, in characteristic $0$ all indecomposable representations are irreducible). Ideally, such a reference would enumerate the representations and offer details on how they are constructed. I already know that there is exactly one irreducible representation $V$ for each dimension $1 \leq \dim V < p$ and all other irreducible representations are of dimension $p$.

Answer 1

I believe those with $p$-character non-zero are irreducible, i.e. the representations of the reduced enveloping algebra $u_\chi(\mathfrak{sl}_2)$ with $\chi\neq 0$ are completely reducible.

For those in the principal block, I think I recall that they are either the differentials of Weyl modules or dual Weyl modules for the algebraic group $SL_2$ or they are projective indecomposable. This was first done in [Premet, The Green ring of a simple three-dimensional Lie p-algebra. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 1991, no. 10, 56–67; translation in Soviet Math. (Iz. VUZ) 35 (1991), no. 10, 51–60 17B10 (17B50)]. But you can also look here [Chari, Vyjayanthi; Premet, Alexander Indecomposable restricted representations of quantum sl2. Publ. Res. Inst. Math. Sci. 30 (1994), no. 2, 335–352. ].

FWIW, the projective modules are easy to describe. The irreducible Steinberg module of dimension $p$ and high weight $p-1$ is projective. Otherwise they look like $A/(B+B)/A$ i.e. simple head and socle isomorphic to $A$ and a direct sum of two isomorphic copies of $B$ in the middle. Moreover if $A=L(r)$ and $B=L(s)$ are simple modules for $SL_2$ with high weights $r$ and $s$, then $r+s=p-2$. One can see this module as the differential of a tilting module $T(2p-2-r)$ for the algebraic group, with high weight $2p-2-r$. We have $T(2p-2-r)$ is uniserial with factors $L(r),L(2p-2-r)\cong L(p-2-r)\otimes L(1)^{(1)}, L(r)$.

If I recall correctly, the Weyl and dual Weyl modules differentiate to modules for $\mathfrak{sl}_2$ which have isotypic socle, isomorphic to a direct sum of simple modules $L(r)$ of high weight $0\leq r<p-1$. Modulo the socle, you get the isotypic head, isomorphic to a direct sum of simple modules $L(s)=L(p-2-r)$. They are configured in a zig-zag fashion so

$$\begin{array}{c c c c c c} L(s)& & L(s) && L(s)\\ &\diagdown\diagup & &\diagdown\diagup &&\diagdown \\ &L(r)&&L(r)&&L(r)\end{array}$$