Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections

2016/02/29 by Valery Gritsenko, Viacheslav V. Nikulin · 34 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebra over a field #Algebraic number #Algebraic structures and combinatorial models #Excellence #Law #Mathematics #Political science #Pure mathematics #math.AG #math.NT #math.QA

paper · pdf · doi:10.1112/plms.12084

published in Proceedings of the London Mathematical Society 116(3), 485-533 (Wiley) · Var2: 75 pages, 16 figures. The exposition polished, some remarks and references added

arxiv created 2016/03/29 · openalex publication_date 2017/11/28 · arxiv updated 2018/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We describe a new large class of Lorentzian Kac–Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices S with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac–Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct Lorentzian Kac–Moody algebras which give their automorphic corrections: they are graded by the S, have the same simple real roots, but their denominator identities are given by automorphic forms with 2-reflective divisors. We give exact constructions of these automorphic forms as Borcherds products and, in some cases, as additive Jacobi liftings.

Citations

• A lecture on Kac--Moody Lie algebras of the arithmetic type
• Reflective modular forms in algebraic geometry
• The Remark on Discriminants of K3 Surfaces Moduli as Sets of Zeros of Automorphic Forms
• The Kodaira dimension of the moduli of K3 surfaces
• Monstrous moonshine and monstrous Lie superalgebras
• Automorphic forms onO s +2,2(R) and infinite products
• Finiteness of 2-reflective lattices of signature (2,n)
• Lie superalgebras
• Sphere Packings, Lattices and Groups