Construction par dualité des algèbres de Kac–Moody symétrisables

Résumé We know that there is a one to one correspondence between Kac–Moody algebras and generalized Cartan matrices. In Kac (“Infinite-Dimensional Lie algebras,” 3rd ed., Cambridge Univ. Press, Cambridge, UK, 1990), one can find a way to reconstruct such an algebra as a Lie algebra presented by generators and relations. The aim of the present work is to give another way to reconstruct those algebras when the Cartan matrix is symmetrisable. Our method will use a semi-classical version of techniques of quantum groups.