In algebra, an integrable module (or integrable representation) of a Kac–Moody algebra g {\displaystyle {\mathfrak {g}}} (a certain infinite-dimensional Lie algebra) is a representation of g {\displaystyle {\mathfrak {g}}} such that (1) it is a sum of weight spaces and (2) 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.
Notes
- Kac, Victor (1990). Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. ISBN 0-521-46693-8.
References
Kac 1990, § 3.6. - Kac, Victor (1990). Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. ISBN 0-521-46693-8. https://books.google.com/books?id=kuEjSb9teJwC&q=Victor%20G.%20Kac&pg=PP1 ↩
Kac 1990, Lemma 3.5. - Kac, Victor (1990). Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. ISBN 0-521-46693-8. https://books.google.com/books?id=kuEjSb9teJwC&q=Victor%20G.%20Kac&pg=PP1 ↩