$\mathcal{U}(\mathfrak{h})$-free modules and coherent families

TL;DR

This paper classifies and constructs U(h)-free modules for Lie algebras of type A and C, introducing new simple modules and demonstrating how translation functors generate higher-rank modules.

Contribution

It establishes the existence conditions for U(h)-free modules, classifies rank 1 modules in type C, and uses translation functors to build higher-rank modules.

Findings

U(h)-free modules only exist for type A and C Lie algebras.

Explicit classification of rank 1 U(h)-free modules in type C.

Construction of new simple sp(2n)-modules.

Abstract

We investigate the category of U(h)-free g-modules. Using a functor from this category to the category of coherent families, we show that U(h)-free modules only can exist when g is of type A or C. We then proceed to classify isomorphism classes of U(h)-free modules of rank 1 in type C, which includes an explicit construction of new simple sp(2n)-modules. Finally, we show how translation functors can be used to obtain simple U(h)-free modules of higher rank.