Classical Lie Algebras at Infinity

by Ivan Penkov, Crystal Hoyt

Overview

Originating from graduate topics courses given by the first author, this book functions as a unique text-monograph hybrid that bridges a traditional graduate course to research level representation theory. The exposition includes an introduction to the subject, some highlights of the theory and recent results in the field, and is therefore appropriate for advanced graduate students entering the field as well as research mathematicians wishing to expand their knowledge. The mathematical background required varies from chapter to chapter, but a standard course on Lie algebras and their representations, along with some knowledge of homological algebra, is necessary. Basic algebraic geometry and sheaf cohomology are needed for Chapter 10. Exercises of various levels of difficulty are interlaced throughout the text to add depth to topical comprehension.

The unifying theme of this book is the structure and representation theory of infinite-dimensional locally reductive Lie algebras and superalgebras. Chapters 1-6 are foundational; each of the last 4 chapters presents a self-contained study of a specialized topic within the larger field. Lie superalgebras and flag supermanifolds are discussed in Chapters 3, 7, and 10, and may be skipped by the reader.

Product Details

ISBN-13:	9783030896591
Publisher:	Springer International Publishing
Publication date:	01/06/2022
Series:	Springer Monographs in Mathematics
Edition description:	1st ed. 2022
Pages:	239
Product dimensions:	6.10(w) x 9.25(h) x (d)

Table of Contents

Preface.- Notation and Terminology. - I. Structure of Locally Reductive Lie Algebras.- 1. Finite-dimensional Lie algebras.- 2. Finite-dimensional Lie superalgebras.- 3. Root-reductive Lie algebras.- 4. Two generalizations.- 5. Splitting Borel subalgebras of sl(infinity), frak o (infinity), sp(infinity) and generalized flags.- 6. General Cartan, Borel and parabolic subalgebras of gl(infinity) and sl(infinity).- II. Modules over Locally Reductive Lie Algebras.- 7. Tensor modules of sl(infinity), frak o(infinity), sp (infinity).- 8. Weight modules.- 9.Generalized Harish-Chandra modules.- III. Geometric aspects. - 10.The Bott-Borel-Weil Theorem.- References.- Index of Notation.- Index.

Editorial Reviews

The book is self-contained in an introductory way, it would be better for the reader to know the basics of Lie algebras and their representations. … the book is most suitable for advanced graduate students and research mathematicians.” (Chengkang Xu, Mathematical Reviews, October, 2022)

“Classical Lie Algebras at Infinity is a very interesting book by I. Penkov and C. Hoyt. … Plenty of exercises of various levels of difficulty and open problems are given throughout the book. This exposition is an invitation to a wide open area of research, aiming to invoke a complex combination of advanced mathematical ideas.” (Mee Seong Im, zbMATH 1490.17025, 2022)

“The book is designed to not only serve as a reference but also to introduce well-prepared people to the subject and its current areas of inquiry. … it seems fair to say that this book will appeal to a somewhat narrower subset of readers of this column than do many of the books reviewed here. But the people in this subset should find this a useful, in the authors’ words, ‘invitation to a wide open research area’.” (Mark Hunacek, MAA Reviews, August 1, 2022)

From the Publisher