Lie Algebras (Dover Books on Mathematics) by Nathan Jacobson (PDF)

Description

Lie group theory, developed by M. Sophus Lie in the nineteenth century, ranks among the more important developments in modern mathematics. Lie algebras comprise a significant part of Lie group theory and are being actively studied today. This book, by Professor Nathan Jacobson of Yale, is the definitive treatment of the subject and can be used as a text for graduate courses.Chapter 1 introduces basic concepts that are necessary for an understanding of structure theory, while the following three chapters present the theory itself: solvable and nilpotent Lie algebras, Cartan’s criterion and its consequences, and split semi-simple Lie algebras. Chapter 5, on universal enveloping algebras, provides the abstract concepts underlying representation theory. The basic results on representation theory are given in three succeeding chapters: the theorem of Ado-Iwasawa, classification of irreducible modules, and characters of the irreducible modules. In Chapter 9 the automorphisms of semi-simple Lie algebras over an algebraically closed field of characteristic zero are determined. These results are applied in Chapter 10 to the problems of sorting out the simple Lie algebras over an arbitrary field. The reader, to fully benefit from this tenth chapter, should have some knowledge about the notions of Galois theory and some of the results of the Wedderburn structure theory of associative algebras.Nathan Jacobson, presently Henry Ford II Professor of Mathematics at Yale University, is a well-known authority in the field of abstract algebra. His book, Lie Algebras, is a classic handbook both for researchers and students. Though it presupposes knowledge of linear algebra, it is not overly theoretical and can be readily used for self-study.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐If you’d like to read a leisurely introduction to Lie Algebras, this is not the text for you. It’s not that the content is particularly difficult, it’s just that Jacobson’s notation (operators acting on the right, fraktur letters, etc.) combined with an antiquated approach to mathematical prose and also a lack of LaTeX formatting lead this otherwise nice book to be much more difficult to parse than intended. Jacobson’s other texts, such as Basic Algebra 1 and 2, are by comparison much easier to read and understand. He offers many exercises which range from trivial to short (but not necessarily easy) proofs and also longer (much longer) computations. In particular, finding a base for the root system E7, which involves pages and pages and pages of notes.I wouldn’t recommend this book to anyone, really. I find math texts from before the 80s or so are often a lot more difficult to read than they feel like they should be.

⭐2

⭐Unfortunately I have found some topics the author was trying to discuss not entirely clear and concise, some definitions I personally found vague. Mr. Jacobson should be more clear when discussing certain topics in the text. An example is Chapter 1 Section 5 on Representations and Modules; I found some ways where he was trying to explain not specific enough as eg xa=xA in defining a right module of the representation space. However, the book was exactly concise and clear in other topics; but this particular section I personally found frustrating

⭐I received the item exactly as described.

⭐Lie groups and Lie algebras have become essential to many parts of mathematics and theoretical physics, with Lie algebras a central object of interest in their own right. This book is a very well thought out and well-written introduction to Lie algebras and it provides an excellent entry point into Lie theory for advanced undergraduates and early graduate students interested in learning about the subject.

⭐I am a research mathematician and writing a research monograph in differential geometry. I found this an excellent reference for filing in the details from Wikipedia. It is a classic in the subject that somehow I had not managed to pick up in my career and it was nice to find an inexpensive copy of this to put on my shelf.

⭐This day in age, it helps having mathematical definitions with well constructed proofs, a suitable glossary, and arguably concrete analogues to abstract forms. Some portions of this book I feel should be revised as it is published material.

⭐This book is a classic book in Lie theory and can be used to be a basic materials for Lie algebria and further infinite Lie algebria. It is also a nice textbook.

⭐Authoritative and comprehensive textbook on Lie Algebras.

⭐The book merits aside, the Kindle edition is so full of errors it’s barely readable. I guess the Kindle edition was ocr scanned from the paper one

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