4 problems

Direct-summand conjecture. For $0<|m|\leq 2$, the maximal submodule $U(m)$ is $\{0\}$, $V(m)=V^{\Lambda_m}$ is a direct summand of ${\mathcal F}ib(m)$, and

Similarity conjecture. The properties of the almost affine Lie superalgebras considered here should be close to those of the corresponding hyperbolic Lie algebras, just as the prop…

Racah–Speiser recursion conjecture. For $m=\pm1,\pm2$, the weights of $P(V^{\Lambda_m}_-)$ follow

Complete reducibility conjecture. For each $m\in{\mathbb Z}$, ${\mathcal F}ib(m)$ completely reduces into a direct sum of one trivial module on level $0$, one non-standard module o…