What is Integrable module?

In algebra, an integrable module of a Kac–Moody algebra g {\displaystyle {\mathfrak {g}}} is a representation of g {\displaystyle {\mathfrak {g}}} such that it is a sum of weight spaces and the Chevalley generators e i , f i {\displaystyle e_{i},f_{i}} of g {\displaystyle {\mathfrak {g}}} are locally nilpotent. For example, the adjoint representation of a Kac–Moody algebra is integrable.