Classical Lie Algebras at Infinity (Springer Monographs in Mathematics)

✍ Scribed by Ivan Penkov, Crystal Hoyt

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 245
• Edition: 1st ed. 2022
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Preface
Acknowledgments
Contents
Notation and Terminology
Part I Structure of Locally Reductive Lie Algebras
1 Finite-Dimensional Lie Algebras
1.1 Lie Algebras
1.2 Cartan Subalgebras and Root Systems
1.3 Borel Subalgebras and Sets of Simple Roots
2 Finite-Dimensional Lie Superalgebras
2.1 Lie Superalgebras
2.2 Classical Lie Superalgebras
2.2.1 Definitions
The General Linear Lie Superalgebra gl(V)
The Orthosymplectic Lie Superalgebra osp(V)
The Lie Superalgebra p(V)
The Lie Superalgebra q(V)
Classical Lie Superalgebras
2.2.2 Invariant Bilinear Forms
2.3 Simple Lie Superalgebras
2.4 Semisimple Lie Superalgebras
2.5 Cartan Subsuperalgebras and Root Systems
2.6 Borel Subsuperalgebras and Sets of Simple Roots
3 Root-Reductive Lie Algebras
3.1 Definitions
3.2 Locally Simple Root-Reductive Lie Algebras
3.3 Description of Root-Reductive Lie Algebras
3.4 Generalized Root Decomposition
3.5 Characterization of Root-Reductive Lie Algebras
4 Two Generalizations
4.1 Root-Reductive Lie Superalgebras
4.1.1 Definitions
4.1.2 Classically Semisimple Lie Superalgebras
4.1.3 The Finite-Dimensional Case
4.1.4 Classical Locally Simple Lie Superalgebras
4.1.5 Description of Classically SemisimpleLie Superalgebras
4.2 Locally Simple Lie Algebras
4.2.1 Diagonal Locally Simple Lie Algebras
4.2.2 Supernatural Numbers
4.2.3 Non-diagonal Locally Simple Lie Algebras
5 Splitting Borel Subalgebras of sl (∞), o(∞), sp(∞) and Generalized Flags
5.1 Splitting Borel Subalgebras
5.1.1 Splitting Borel Subalgebras of sl (∞)
5.1.2 Splitting Borel Subalgebras of o(∞) and sp(∞)
5.2 Generalized Flags
5.2.1 Finite-Dimensional Flags
5.2.2 Generalized Flags
5.2.3 Maximal Chains and Parabolic Subalgebras of sl(∞)
5.2.4 The Case of o(∞) and sp(∞)
6 General Cartan, Borel and Parabolic Subalgebras of gl (∞) and sl(∞)
6.1 Cartan Subalgebras of sl (∞) and gl(∞)
6.1.1 Toral Subalgebras
6.1.2 Dual Systems
6.1.3 Cartan Subalgebras
6.2 Borel and Parabolic Subalgebras of sl(∞) and gl(∞)
6.3 Closing Remarks
Part II Modules over Locally Reductive Lie Algebras
7 Tensor Modules of sl(∞), o(∞), sp(∞)
7.1 Structure of Mixed Tensor Products
7.2 The Category Tg
7.3 Injective Objects in Tg
7.4 Vanishing Theorem for Ext in Tg
7.5 Koszulity of the Category Tg
7.6 Koszul Self-duality of the Ring Asl(∞)
7.7 Equivalence of the Categories To(∞) and Tsp(∞)
7.8 Categories of Tensor Modules for Lie Superalgebras
7.8.1 The Case of g=gl(∞|∞)
7.8.2 The Case of g=osp(∞|∞)
7.9 Further Developments
8 Weight Modules
8.1 Preliminaries
8.2 Weight Modules for Finite-Dimensional Lie Algebras
8.2.1 The h-Support of a (g,h)-Weight Module
8.2.2 The Shadow Decomposition
8.2.3 (g,h)-Modules in a Special Case
8.2.4 General (g,h)-Modules
8.2.5 Fundamental Results for Reductive g
8.2.6 Cuspidal Modules and Mathieu's Coherent Families
8.2.7 Weight Modules for Lie Superalgebras
8.3 Weight Modules for Root-Reductive Lie Algebras
8.3.1 Main Differences from the Finite-Dimensional Case
8.3.2 The Shadow Decomposition
8.3.3 Simple Bounded Weight Modules of sl(∞), o(∞), sp(∞)
8.3.4 Simple Highest Weight Modules of sl(∞), o(∞), sp(∞) with Finite-Dimensional Weight Spaces
8.4 Categories of Modules over Root-Reductive Lie Algebras
8.4.1 Categories of Integrable Modules
8.4.2 Categories Containing Non-Integrable Modules
8.5 Closing Remarks
9 Generalized Harish-Chandra Modules
9.1 Introduction
9.2 The Case where g is Finite Dimensional
9.2.1 Construction of (g,k)-Modules
9.2.2 Reconstruction Theorems
9.2.3 Fernando–Kac Subalgebra of a Fundamental Series Module
9.2.4 The Case ksl(2)
9.3 The Case when g is Infinite Dimensional
9.4 Closing Remarks
Part III Geometric Aspects
10 The Bott–Borel–Weil Theorem
10.1 The Classical Bott–Borel–Weil Theorem
10.1.1 Introductory Considerations
10.1.2 The Main Result
10.2 Toward a Bott–Borel–Weil Theorem for Lie Supergroups
10.2.1 Preliminaries on Lie Supergroups and Flag Supermanifolds
10.2.2 Bott–Borel–Weil Theorem for Typical Weights
10.3 Toward a Bott-Borel-Weil Theorem for Locally Reductive Ind-Groups
10.3.1 The Case of a Line Bundle
10.3.2 The Case of Growing Rank
10.4 Open Problems
References
Index of Notation
Index