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