Quantum analogues of a coherent family of modules at roots of unity: A3

Abstract

To a given coherent family of virtual representations of a complex semiesimple Lie algebra we associate in [P] a coherent family of virtual representations of the corresponding quantum group at roots of unity[P, section 2]. This is recalled fairly explicitly in section 2 below. We also proposed a conjecture there that under some hyptoheses the members of the family in a certain positive cone are actually modules (as opposed to a 'virtual' module which is in general only a difference of two modules). We verify the validity of this conjecture for A₂ and B₂ But first we recall in some length the ideas in [P] without detailed proofs.