Algebra and Applications 1

Non-associative Algebras and Categories

Abdenacer Makhlouf

• Publisher: Wiley-ISTE
• Year: 2021
• ISBN:
  9781119818151
• Edition: 1
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

This book is part of Algebra and Geometry, a subject within the SCIENCES collection published by ISTE and Wiley, and the first of three volumes specifically focusing on algebra and its applications. Algebra and Applications 1 centers on non-associative algebras and includes an introduction to derived categories. The chapters are written by recognized experts in the field, providing insight into new trends, as well as a comprehensive introduction to the theory.

The book incorporates self-contained surveys with the main results, applications and perspectives. The chapters in this volume cover a wide variety of algebraic structures and their related topics. Jordan superalgebras, Lie algebras, composition algebras, graded division algebras, non-associative C*- algebras, H*-algebras, Krichever-Novikov type algebras, preLie algebras and related structures, geometric structures on 3-Lie algebras and derived categories are all explored. Algebra and Applications 1 is of great interest to graduate students and researchers.

Each chapter combines some of the features of both a graduate level textbook and of research level surveys.

1
Jordan Superalgebras

Consuelo MARTINEZ¹ and Efim ZELMANOV²

¹Department of Mathematics, University of Oviedo, Spain

²Department of Mathematics, University of California San Diego, USA

1.1. Introduction

Superalgebras appeared in a physical context in order to study, in a unified way, supersymmetry of elementary particles. Jordan algebras grew out of quantum mechanics and gained prominence due to their connections to Lie theory. In this chapter, we survey Jordan superalgebras focusing on their connections to other subjects. In this section we introduce some basic definitions and in section 1.2 we give the Tits–Kantor–Koecher construction that shows the way in which Lie and Jordan structures are connected. In section 1.3, we show examples of some basic superalgebras (the so-called classical superalgebras). Section 1.4 is about the notion of brackets and explains how to construct superalgebras using different types of brackets. Section 1.5 explains Cheng–Kac superalgebras, an important class of superalgebras that appeared for the first time in the context of superconformal algebras. The classification of Jordan superalgebras is explained in section 1.6, and it includes the cases of an algebraically closed field of zero characteristics, the case of prime characteristic, both for Jordan superalgebras with semisimple even part and with non-semisimple even part, and the case of non-unital Jordan superalgebras. Finally, in section 1.7, we give some general ideas about Jordan superconformal algebras. Throughout the chapter, all algebras are considered over a field F, charF ≠ 2.

DEFINITION 1.1.– A (linear) Jordan algebra is a vector space J with a ...