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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.

In this chapter we define Borcherds-Kac-Moody Lie superalgebras and explain what their structure is. We will abbreviate this name to BKM. We will take C to be the base field instead of R as in [Borc4].

In this chapter we introduce vector valued modular forms and show how to derive their theta transforms. As we concentrate only on the properties needed for our classification purpose given in sections 5.2 and 5.3, for an in depth study of these objects, the reader can consult the following books: [Borc11], [EicZ], [Frei], [Mi], [Shi], [Serr2].

The aim of this chapter is to present the aspects of the theory necessary for the construction of BKM superalgebras. The purpose of this book is not the study of Γ-graded vertex algebras per se, so we only give the details needed for our purpose. Hence the exposition does not pretend to be a complete treatment of Γ-graded vertex algebras - a vast subject with a rich literature dedicated to it.

The aim of the chapter is to give the reader the tools to understand the classification of the “interesting” BKM superalgebras. This work is still in progress today and has not been completed.