Continuum Kac–Moody Algebras

Abstract

We introduce a new class of infinite-dimensional Lie algebras, which arise as continuum colimits of Borcherds-Kac-Moody algebras. They are associated with a topological generalization of the notion of quiver, where vertices are replaced by intervals in a real one-dimensional topological space, and are described by a continuum root system with no simple root. For these Lie algebras, we prove an analogue of the Gabber-Kac-Serre theorem, providing a complete set of defining relations featuring only quadratic Serre relations.