Integrable module

In algebra, an integrable module (or integrable representation) of a Kac–Moody algebra ${\displaystyle {\mathfrak {g}}}$ (a certain infinite-dimensional Lie algebra) is a representation of ${\displaystyle {\mathfrak {g}}}$ such that (1) it is a sum of weight spaces and (2) the Chevalley generators ${\displaystyle e_{i},f_{i}}$ of ${\displaystyle {\mathfrak {g}}}$ are locally nilpotent.¹ For example, the adjoint representation of a Kac–Moody algebra is integrable.²