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

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.

Var2: 75 pages, 16 figures. The exposition polished, some remarks and references added

References in corpus (3)

• The Kodaira dimension of the moduli of K3 surfaces
• Reflective modular forms in algebraic geometry
• Classification of log del Pezzo surfaces of index $\le 2$

Cited by in corpus (14)

• On some free algebras of orthogonal modular forms
• The classification of free algebras of orthogonal modular forms
• The D_8-tower of weak Jacobi forms and applications
• The classification of 2-reflective modular forms
• Quasi-pullback of Borcherds products
• Projective spaces as orthogonal modular varieties
• Simple lattices and free algebras of modular forms
• Antisymmetric paramodular forms of weight 3
• Natural Construction of Ten Borcherds-Kac-Moody Algebras Associated with Elements in $M_{23}$
• Weyl invariant $E_8$ Jacobi forms
• Reflective modular forms on lattices of prime level
• Fano Shimura varieties with mostly branched cusps
• Reflective modular forms: A Jacobi forms approach
• Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac--Moody algebras