Lie Algebras and Locally Compact Groups by Irving Kaplansky – A Comprehensive Mathematics Reference on Semi-Simple Lie A

Introduction

For serious students of advanced mathematics, few works carry the weight and clarity of Irving Kaplansky's classic text. Lie Algebras and Locally Compact Groups is a rigorous yet accessible exploration of two deeply interconnected fields: the algebraic structure of Lie algebras and the topological resolution of Hilbert's fifth problem. Originally developed from Kaplansky's own lecture notes at the University of Chicago, this hardcover edition from the University of Chicago Press remains an indispensable resource for graduate students and researchers in pure mathematics. Whether you are delving into the intricacies of semi-simple Lie algebras or tracing the historical path to the solution of a famous Hilbert problem, this book offers a masterful blend of theory and insight.

Book Overview

This volume is divided into two distinct but complementary parts. The first chapter, titled Lie Algebras, presents a streamlined treatment of the structure theory of semi-simple Lie algebras in characteristic zero, following the classical approaches of Killing and Cartan. The second chapter, The Structure of Locally Compact Groups, addresses the celebrated solution of Hilbert's fifth problem as established by Gleason, Montgomery, and Zippin in 1952. Together, these sections provide a coherent journey from algebraic foundations to topological applications, making the book a unique addition to any mathematician's library.

Key Highlights

• Classic exposition by a renowned mathematician, Irving Kaplansky
• Connects Lie algebra theory with the solution of Hilbert's fifth problem
• Based on authentic lecture notes, preserving a conversational yet precise tone
• Hardcover edition from the prestigious University of Chicago Press
• Ideal for graduate-level courses and self-study in algebra and topology

Inside the Book

Kaplansky begins with a concise introduction to the fundamentals of Lie algebras, moving quickly into the structure theory of semi-simple algebras. Readers will encounter the Cartan subalgebra, root systems, and the classification theorems that form the backbone of the subject. The second half shifts focus to locally compact groups, where the author carefully unpacks the topological conditions that lead to the resolution of Hilbert's fifth problem. The text is dense with theorems, proofs, and explanatory remarks that reflect Kaplansky's gift for clarity. No extraneous material distracts from the core mathematics.

Key Topics

• Definition and basic properties of Lie algebras
• Semi-simple Lie algebras in characteristic zero
• Killing form and Cartan's criteria
• Root systems and Dynkin diagrams
• Locally compact groups and their structure
• Hilbert's fifth problem and its solution
• Gleason, Montgomery, and Zippin's contributions

Reader Benefits

This book offers more than just mathematical facts; it provides a window into the thinking of one of the twentieth century's great algebraists. By working through Kaplansky's arguments, readers develop a deeper understanding of how algebraic and topological ideas converge. The lecture-note format makes the material feel immediate and alive, as if sitting in Kaplansky's classroom. For Indian students preparing for advanced research or competitive examinations in mathematics, this text builds a solid foundation in two critical areas of modern mathematics.

Learning Outcomes

• Gain a thorough command of the structure theory of semi-simple Lie algebras
• Understand the role of the Killing form and Cartan subalgebras
• Learn the key steps in the solution of Hilbert's fifth problem
• Develop the ability to read and construct rigorous proofs in algebra and topology
• Bridge the gap between abstract algebra and topological group theory

Who Should Read

This book is intended for graduate students in mathematics, especially those specializing in algebra, topology, or Lie theory. Researchers and faculty members seeking a concise reference on semi-simple Lie algebras or the structure of locally compact groups will find it invaluable. Advanced undergraduates with a strong background in abstract algebra and point-set topology can also benefit, provided they are willing to engage with challenging material. It is not a beginner's text, but rather a focused monograph for those ready to explore deep mathematical landscapes.

About the Author

Irving Kaplansky (1917–2006) was a Canadian-American mathematician known for his profound contributions to algebra, ring theory, and functional analysis. He spent much of his career at the University of Chicago, where he served as a professor and later as director of the Mathematical Sciences Research Institute. Kaplansky's expository style is celebrated for its elegance and clarity, and his books remain classics in their fields. His work earned him numerous honors, including the Leroy P. Steele Prize for Lifetime Achievement.

About the Publisher

The University of Chicago Press has been a leading academic publisher since 1890. Renowned for its rigorous editorial standards and commitment to scholarly excellence, the Press has published seminal works in mathematics, science, and the humanities. This hardcover edition reflects the Press's tradition of producing durable, beautifully crafted books that endure on library shelves and in the hands of serious readers.

Conclusion

Lie Algebras and Locally Compact Groups is more than a textbook; it is a mathematical treasure. Kaplansky's lucid exposition, combined with the prestige of the University of Chicago Press, makes this volume a must-have for anyone serious about advanced mathematics. Whether you are studying for a graduate course, preparing for research, or simply seeking to understand two of the most elegant theories in modern mathematics, this book will guide you with authority and grace. Add this classic to your collection today from Bookshops.in.