Lie superalgebras and enveloping algebras (Q2882522)

This book is intended for those who know some classical Lie theory and who wish to learn about Lie superalgebras, that is, \(\mathbb{Z}_{2}\)-graded Lie algebras. It develops the theory of Lie superalgebras from the start, going on to discuss constructions, classifications, enveloping algebras and the corresponding methods from ring theory, representation theory and cohomology.NEWLINENEWLINEThe author's stated intention is to gather together ``what is known about Lie superalgebras and their representations'' and the nearly 500 pages of this volume demonstrate that indeed a lot is known. A second stated aim is to collect results that have to date only appeared in research journals, for the convenience of future researchers in the area. (Those new to this topic should be reassured that this does not mean that key background is omitted -- to the contrary, this is provided in a very thorough way.)NEWLINENEWLINEWe now summarise the contents. The book begins with five chapters on basic properties, including definitions, the statement of the Classification Theorem and constructions of the classical simple Lie superalgebras. Chapter 6 introduces the enveloping algebra and proves the PBW basis theorem for it, while Chapter 7 discusses some methods from ring theory used later on. Chapter 8 begins the representation theory, which continues in Chapters 9 and 10, which discuss Verma modules. Chapter 11 concerns Schur--Weyl duality for the general linear Lie superalgebras and some others and Chapter 12 introduces supersymmetric polynomials. Chapter 13 deals with the Harish-Chandra homomorphism and central characters, while Chapter 13 concerns finite dimensional representations with a particular focus on the orthosymplectic case. Chapter 15 returns to enveloping algebras and problems concerning the prime and primitive ideals therein. Chapter 16 describes the cohomology of Lie superalgebras and Chapter 17 covers zero divisors in the enveloping algebras. The book concludes with Chapter 18 on affine Lie superalgebras, followed by two appendices. The first covers (necessarily very briefly) Lie theory, Hopf algebras and the Diamond Lemma. The second provides some tables of data relating to the root systems of Lie superalgebras of small rank.NEWLINENEWLINEThere are a large number of exercises provided throughout the text and suggestions for using the book as a course text may be found in the Preface.NEWLINENEWLINEIt seems to this reviewer that he has been well-chosen, as part of the natural target audience for this book. As such, I have indeed found it to be accessible and interesting and it contains much that I look forward to considering in greater depth in the future. It is clear that this text will be very valuable to all who are new to the ``super world'' and it makes a very good case for its contents being better known and for the continued development of the topic. Those who take up the latter challenge will find this book a fine companion.