682 research outputs found
Lattices with many Borcherds products
We prove that there are only finitely many isometry classes of even lattices
of signature for which the space of cusp forms of weight
for the Weil representation of the discriminant group of is trivial. We
compute the list of these lattices. They have the property that every Heegner
divisor for the orthogonal group of can be realized as the divisor of a
Borcherds product. We obtain similar classification results in greater
generality for finite quadratic modules.Comment: 31 pages, 1 figur
Introduction to Vertex Algebras, Borcherds Algebras, and the Monster Lie Algebra
The theory of vertex algebras constitutes a mathematically rigorous axiomatic
formulation of the algebraic origins of conformal field theory. In this context
Borcherds algebras arise as certain ``physical'' subspaces of vertex algebras.
The aim of this review is to give a pedagogical introduction into this
rapidly-developing area of mathemat% ics. Based on the machinery of formal
calculus we present the axiomatic definition of vertex algebras. We discuss the
connection with conformal field theory by deriving important implications of
these axioms. In particular, many explicit calculations are presented to stress
the eminent role of the Jacobi identity axiom for vertex algebras. As a class
of concrete examples the vertex algebras associated with even lattices are
constructed and it is shown in detail how affine Lie algebras and the fake
Monster Lie algebra naturally appear. This leads us to the abstract definition
of Borcherds algebras as generalized Kac-Moody algebras and their basic
properties. Finally, the results about the simplest generic Borcherds algebras
are analysed from the point of view of symmetry in quantum theory and the
construction of the Monster Lie algebra is sketched.Comment: 55 pages, (two minor changes thanks to comment by R. Borcherds
Standard and Non-standard Extensions of Lie algebras
We study the problem of quadruple extensions of simple Lie algebras. We find
that, adding a new simple root , it is not possible to have an
extended Kac-Moody algebra described by a Dynkin-Kac diagram with simple links
and no loops between the dots, while it is possible if is a
Borcherds imaginary simple root. We also comment on the root lattices of these
new algebras. The folding procedure is applied to the simply-laced triple
extended Lie algebras, obtaining all the non-simply laced ones. Non- standard
extension procedures for a class of Lie algebras are proposed. It is shown that
the 2-extensions of , with a dot simply linked to the Dynkin-Kac diagram
of , are rank 10 subalgebras of . Finally the simple root
systems of a set of rank 11 subalgebras of , containing as sub-algebra
, are explicitly written.Comment: Revised version. Inaccurate statements corrected. Expanded version
with added reference
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