Quantum analogues of a coherent family of modules at roots of unity: A 3

To a given coherent family of virtual representations of a complex semiesimple Lie algebra we associate elsewhere a coherent family of virtual representations of the corresponding quantum group at roots of unity. We also proposed a conjecture there that under some hyptoheses the members of the family in a certain positive cone are actually modules (and not just a virtual module, i.e., a difference of two modules). The validity of this conjecture for A ₂ and B ₂ has been verified by the author. In this paper we solve the problem for A ₃.