Oxidizing Borcherds symmetries

Notes on Borcherds Products

These are the notes for a short course on Borcherds Products held in Aachen on 1st2nd August 2012. The lectures are intended for someone who does not know what a Borcherds product is, but does know what a modular form is. In preparing these notes I’ve used the following sources: [1, 2, 3, 4, 5]. All the correct statements in these notes are taken from those sources, and all the mistakes are my ...

Local Borcherds products

Generalized Moonshine Ii: Borcherds Products

The goal of this paper is to construct infinite dimensional Lie algebras using infinite product identities, and to use these Lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and Lie algebras. The Lie algebras that we construct conjecturally appear in an orbifold conformal field theory with symmetries given by the monster ...

Canonical Bases of Borcherds - Cartan

We study the canonical basis for the negative part U− of the quantum generalized Kac-Moody algebra associated to a symmetric BorcherdsCartan matrix. The algebras U− associated to two different matrices satisfying certain conditions may coincide (6.3). We show that the canonical bases coincide provided that the algebras U− coincide (Theorem 6.3.5). We also answer partially a question by Lusztig ...

Lattices with many Borcherds products

We prove that there are only finitely many isometry classes of even lattices L of signature (2, n) for which the space of cusp forms of weight 1 + n/2 for the Weil representation of the discriminant group of L is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of L can be realized as the divisor of a Borcherds product. W...