Automorphic Forms and Lie Superalgebras
Description
A principal ingredient in the proof of the Moonshine Theorem, connecting the Monster group to modular forms, is the infinite dimensional Lie algebra of physical states of a chiral string on an orbifold of a 26 dimensional torus, called the Monster Lie algebra. It is a Borcherds-Kac-Moody Lie algebra with Lorentzian root lattice; and has an associated automorphic form having a product expansion describing its structure. Lie superalgebras are generalizations of Lie algebras, useful for depicting supersymmetry - the symmetry relating fermions and bosons. Most known examples of Lie superalgebras with a related automorphic form such as the Fake Monster Lie algebra whose reflection group is given by the Leech lattice arise from (super)string theory and can be derived from lattice vertex algebras. The No-Ghost Theorem from dual resonance theory and a conjecture of Berger-Li-Sarnak on the eigenvalues of the hyperbolic Laplacian provide strong evidence that they are of rank at most 26.
The aim of this book is to give the reader the tools to understand the ongoing classification and construction project of this class of Lie superalgebras and is ideal for a graduate course. The necessary background is given within chapters or in appendices.
Reviews / Votes
From the reviews:
"The aim of this book is to present the very active theory of Borcherds-Kac-Moody Lie algebras and of G-graded vertex algebras at a level accessible to graduate students and mathematicians from other fields. An introduction of this type is certainly very desirable . . The author of this review, being a mathematician from a different field . may be a good test person for this book. I like the concept of the book . ." (Rainer Schulze-Pillot, Mathematical Reviews, Issue 2008 e)
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Content
Chapter 1 Introduction (p. 1)
In this book, we give an exposition of the theory of Borcherds-Kac-Moody Lie algebras and of the ongoing classification and explicit construction project of a subclass of these infinite dimensional Lie algebras. We try to keep the material as elementary as possible. More precisely, our aim is to present some of the theory developed by Borcherds to graduate students and mathematicians from other fields.
Some familiarity with complex finite dimensional semisimple Lie algebras, group representation theory, topology, complex analysis, Fourier series and transforms, smooth manifolds, modular forms and the geometry of the upper half plane can only be helpful. However, either in the appendices or within specific chapters, we give the definitions and results from basic mathematics needed to understand the material presented in the book.
We only omit proofs of properties well covered in standard undergraduate and graduate textbooks. There are several excellent reference books on the above subjects and we will not attempt to list them here. However for the purpose of understanding the classification and construction of Borcherds-Kac-Moody (super)algebras the following are particularly useful.
Serre’s approach to the theory of finite dimensional semisimple Lie algebras in [Ser1] is conducive to the construction of Borcherds-Kac-Moody Lie algebras as it emphasizes the presentation of finite dimensional semi-simple Lie algebras via generators and relations. For a first approach to automorphic forms and the geometry of the upper half plane, the book by Shimura may be a place to start at [Shi].
We also do not replicate the proofs of properties of Kac-Moody Lie algebras that are treated in depth in the now classical reference book by Kac [Kac14]. Borcherds-Kac-Moody Lie algebras are generalizations of symmetrizable (see Remarks 2.1.10) Kac-Moody Lie algebras, themselves generalizations of finite dimensional semi-simple Lie algebras. That this further level of generality is needed was first shown in the proof of the moonshine theorem given by Borcherds [Borc7], where the theory of Borcherds-Kac-Moody Lie algebras plays a central role. So let us brie.y explain what this theorem is about.
1.1 The Moonshine Theorem
The remarkable Moonshine Theorem, conjectured by Conway and Norton and one part of which was proved by Frenkel, Lepowsky and Meurman, and another by Borcherds, connects two areas apparently far apart: on the one hand, the Monster simple group and on the other modular forms. Any connection found between an object which has as yet played a limited abstract role and a more fundamental concept is always very fascinating.
Also it is not surprising that ideas on which the proof of such a result are based would give rise to many new questions, thus opening up different research directions and finding applications in a wide variety of areas.
1.1.1 A Brief History
The famous Feit-Thompson Odd order Theorem [FeitT] implying that the only finite simple groups of odd order are the cyclic groups of prime order, sparked off much interest in classifying the inite simple groups in the sixties and seventies.
