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Moscow Mathematical Journal

Volume 22, Issue 2, April–June 2022  pp. 177–224.

Continuum Kac–Moody Algebras

Authors:  Andrea Appel (1), Francesco Sala (2), and Olivier Schiffmann (3)
Author institution:(1) Università di Parma, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Italy
(2) Università di Pisa, Dipartimento di Matematica, Italy;
Kavli IPMU (WPI), UTIAS, The University of Tokyo, Japan
(3) Laboratoire de Mathématiques, Université de Paris-Sud Paris-Saclay, France


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.

2020 Math. Subj. Class. Primary: 17B65; Secondary: 17B67.

Keywords: Continuum quivers, Lie algebras, Borcherds–Kac–Moody algebras.

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