Lie algebras of topological quivers

Andrea Appel, Francesco Sala, Olivier Schiffmann

Published 2018-12-20Version 1

We introduce a new class of infinite-dimensional Lie algebras, which do not fall into the realm of Kac-Moody algebras (or their generalizations by Borcherds and Bozec) but 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 one-dimensional topological space. They display certain exotic features such as a Cartan subalgebra of uncountable dimension, a continuum root system with no simple roots, and they are presented by exclusively quadratic Serre relations. For these Lie algebras, we prove an analogue of the Gabber-Kac-Serre theorem, providing a complete set of defining relations.