The Functor η : mod A → mod C(B)

Since C(B) is the semidirect extension of B by J,
a C(B)-module is a pair (N,γ), where N is a B-module, and γ is a B-module homomorphism
We now define a functor η as follows: Let M be an A-module, then

We have to rewrite J in terms of A: there is a canonical isomorphism

In order to define γ, we just form the corresponding induced exact sequence:

Theorem: η is an equivalence (mod A)/T → (mod C(B))/S.

Corollary: The families of indecomposable C(B)-modules are (nearly bijectively) indexed by the positive roots of the corresponding Kac-Moody Lie-algebra.