Automorphic Forms and Lie Superalgebras
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 alg
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Algebra and Applications Volume 5 Managing Editor: Alain Verschoren University of Antwerp, Belgium Series Editors: Eric Friedlander Northwestern University, U.S.A. John Greenlees Sheffield University, U.K. Gerhard Hiss Aachen University, Germany Ieke Moerdijk Utrecht University, The Netherlands Idun Reiten Norwegian University of Science and Technology, Norway Christoph Schweigert Hamburg University, Germany Mina Teicher Bar-llan University, Israel Algebra and Applications aims to publish well-written and carefully refereed monographs with up-to-date expositions of research in all fields of algebra, including its classical impact on commutative and noncommutative algebraic and differential geometry, K-theory and algebraic topology, and further applications in related domains, such as number theory, homotopy and (co)homology theory through to discrete mathematics and mathematical physics. Particular emphasis will be put on state-of-the-art topics such as rings of differential operators, Lie algebras and super-algebras, group rings and algebras, Kac-Moody theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well as their applications within mathematics and beyond. Books dedicated to computational aspects of these topics will also be welcome.
Automorphic Forms and Lie Superalgebras by
Urmie Ray Université de Reims, Reims, France
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 ISBN-13 ISBN-10 ISBN-13
1-4020-5009-7 (HB) 978-1-4020-5009-1 (HB) 1-4020-5010-0 (e-book) 978-1-4020-5010-7 (e-book)
Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com
Printed on acid-free paper
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To my Mother
Contents Preface
ix
1 Introduction 1.1 The Moonshine Theorem . . . . . . . . . 1.1.1 A Brief History . . . . . . . . . . . 1.1.2 The Theorem . . . . . . . . . . . . 1.2 Borcherds-Kac-Moody Lie Superalgebras . 1.3 Vector Valued Modular Forms . . . . . . . 1.4 Borcherds-Kac-Moody Lie Algebras and Modular forms . . . . . . . . . . . . . 1.5 Γ-graded Vertex Algebras . . . . . . . . . 1.6 A Construction of a Class of Borcherds-Kac-Moody Lie (super)algebras 2 Borcherds-Kac-Moody Lie Superalgebras 2.1 Definitions and Elementary Properties . . 2.2 Bilinear Forms . . . . . . . . . . . . . . . 2.3 The Root System . . . . . . . . . . . . . . 2.4 Uniqueness of the Generalized Cartan Matrix . . . . . . . . . . . . . . . . . . . . 2.5 A Characterization of BKM Superalgebras 2.6 Character and Denominator Formulas . . 3 Singular Theta Transforms of Vector 3.1 Lattices . . . . . . . . . . . . . . . . 3.2
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