Automorphic Forms and Lie Superalgebras (eBook)

Contents 7
Preface 9
1 Introduction 10
1.1 The Moonshine Theorem 11
1.1.1 A Brief History 11
1.1.2 The Theorem 12
1.2 Borcherds-Kac-Moody Lie Superalgebras 14
1.3 Vector Valued Modular Forms 16
1.4 Borcherds-Kac-Moody Lie Algebras and Modular forms 16
1.5 G-graded Vertex Algebras 18
1.6 A Construction of a Class of Borcherds- Kac- Moody Lie ( super)algebras 18
2 Borcherds-Kac-Moody Lie Superalgebras 22
2.1 Definitions and Elementary Properties 22
2.2 Bilinear Forms 35
2.3 The Root System 47
2.4 Uniqueness of the Generalized Cartan Matrix 67
2.5 A Characterization of BKM Superalgebras 75
2.6 Character and Denominator Formulas 81
3 Singular Theta Transforms of Vector Valued Modular Forms 102
3.1 Lattices 102
3.2 Ordinary Modular Functions 104
3.3 Vector Valued Modular Functions 111
3.4 The Singular Theta Correspondence 122
4 G-Graded Vertex Algebras 138
4.1 The Structure of G-graded Vertex Algebras 138
4.2 G-Graded Lattice Vertex Algebras 158
4.3 From Lattice Vertex Algebras to Lie Algebras 176
5 Lorentzian BKM Algebras 186
5.1 Introduction 186
5.2 Automorphic forms on Grassmannians 188
5.3 Vector Valued Modular forms and LBKM Algebras 195
5.4 An Upper Bound for the Rank of the Root Lattices of LBKM Algebras? 216
5.5 A Construction of LBKM Algebras from Lattice Vertex Algebras 222
Appendix A Orientations and Isometry Groups 248
Appendix B Manifolds 250
B.1 Some Elementary Topology 250
B.2 Manifolds 251
B.3 Fibre Bundles and Covering Spaces 256
Appendix C Some Complex Analysis 260
C.1 Measures and Lebesgue Integrals 260
C.2 Complex Functions 262
C.3 Integration 263
C.4 Some Special Functions 264
Appendix D Fourier Series and Transforms 272
Notation 278
Bibliography 282
Index 292

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.