Continuum Kac–Moody algebras

We introduce a new class of infinite-dimensional Lie algebras, which we refer to as continuum Kac–Moody algebras. Their construction is closely related to that of usual Kac–Moody algebras, but they feature a continuum root system with no simple roots. Their Cartan datum encodes the topology of a one-dimensional real space and can be thought of as a generalization of a quiver, where vertices are replaced by connected intervals. 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. Moreover, we provide an alternative realization as continuum colimits of symmetric Borcherds–Kac–Moody algebras with at most isotropic simple roots. The approach we follow deeply relies on the more general notion of a semigroup Lie algebra and its structural properties.