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© ISTE Ltd 2021
The rights of Abdenacer Makhlouf to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2020938694
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78945–017-0
ERC code:
PE1 Mathematics
PE1_2 Algebra
PE1_5 Lie groups, Lie algebras
PE1_12 Mathematical physics
Foreword
Abdenacer MAKHLOUF
IRIMAS-Department of Mathematics, University of Haute Alsace, Mulhouse, France
We set out to compile several volumes pertaining to Algebra and Applications in order to present new research trends in algebra and related topics. The subject of algebra has grown spectacularly over the last several decades; algebra reasoning and combinatorial aspects turn out to be very efficient in solving various problems in different domains. Our objective is to provide an insight into the fast development of new concepts and theories. The chapters encompass surveys of basic theories on non-associative algebras, such as Jordan and Lie theories, using modern tools in addition to more recent algebraic structures, such as Hopf algebras, which are related to quantum groups and mathematical physics.
We provide self-contained chapters on various topics in algebra, each combining some of the features of both a graduate-level textbook and a research-level survey. They include an introduction with motivations and historical remarks, the basic concepts, main results and perspectives. Furthermore, the authors provide comments on the relevance of the results, as well as relations to other results and applications.
This first volume deals with non-associative and graded algebras (Jordan algebras, Lie theory, composition algebras, division algebras, pre-Lie algebras, Krichever–Novikov type algebras, C*-algebras and H*-algebras) and provides an introduction to derived categories.
I would like to express my deep gratitude to all the contributors of this volume and ISTE Ltd for their support.
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 linear binary operation (x, y) ↦ xy satisfying the following identities:
(J1) xy = yx (commutativity);
(J2) (x2y)x = x2(yx) ∀x, y ∈ J (Jordan identity).
Instead of (J2) we can consider the corresponding linearized identity:
(J’2) (xy)(zu) + (xz)(yu) + (xu)(yz) = ((xy)z)u + ((xu)z)y + ((yu)z)x ∀x, y, z, u ∈ J.
REMARK 1.1.– A Lie algebra L is a vector space with a linear binary operation (x, y) ↦ [x, y] satisfying the following identities:
(L1) [x, y] = –[y, x] (anticommutativity);
(L2) [[x, y], z] + [[y, z], x] + [[z, x], y] = 0 for arbitrary elements x, y, z ∈ J (Jacobi identity).
EXAMPLE 1.1.– If A is an associative algebra, then (A(+), ∙), where a ∙ b = ab + ba is a Jordan algebra, and (A(–), [, ]), where [a, b] = ab – ba is a Lie algebra. Both A(+) and A(–) have the same underlying vector space as A.
DEFINITION 1.2.– A superalgebra A is an algebra with a ℤ/2ℤ-grading. So
Elements of
Elements in
Note that