Abstract Lie Algebras (Dover Books on Mathematics) by David J Winter (PDF)

Description

Solid but concise, this account of Lie algebra emphasizes the theory’s simplicity and offers new approaches to major theorems. Author David J. Winter, a Professor of Mathematics at the University of Michigan, also presents a general, extensive treatment of Cartan and related Lie subalgebras over arbitrary fields.Preliminary material covers modules and nonassociate algebras, followed by a compact, self-contained development of the theory of Lie algebras of characteristic 0. Topics include solvable and nilpotent Lie algebras, Cartan subalgebras, and Levi’s radical splitting theorem and the complete reducibility of representations of semisimple Lie algebras. Additional subjects include the isomorphism theorem for semisimple Lie algebras and their irreducible modules, automorphism of Lie algebras, and the conjugacy of Cartan subalgebras and Borel subalgebras. An extensive theory of Cartan and related subalgebras of Lie algebras over arbitrary fields is developed in the final chapter, and an appendix offers background on the Zariski topology.

User’s Reviews

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

⭐The title says it all. This book really is about abstract Lie algebras, not the concrete Lie algebras of Lie groups which are of greatest interest in differential geometry and physics.The publisher’s summary of this book is detailed and technical. (See the back cover, which is reproduced as Amazon’s summary.) This book is at the graduate level. It assumes that you already know the basics of Lie algebras and now want to go more deeply into this subject as a preparation for research. There is no mention of Lie groups, although the original motivation for studying Lie algebras clearly lies in the Lie group context. (The Lie algebra of a Lie group is the algebra of commutators of left invariant vector fields on the group.) An abstract Lie algebra is, technically speaking, a unitary left module over a commutative unitary ring, together with a product operation on the module which satisfies distributivity, anti-commutativity, the Jacobi identity (which implies non-associativity) and scalarity with respect to the ring. In other words, it’s a kind of vector space with an anti-commutative product operation obeying Jacobi’s identity. (The author’s definition is expressed somewhat differently on page 18.)The material in this book is clearly meant for algebraists, not for differential geometers. There’s no analysis here of the kind you would expect in the Lie group context. The questions which are answered here are the questions which algebraists ask. (For example, the section on “classification of split semisimple Lie algebras” gets 41 pages, which is 28.7% of the whole book.) On the other hand, many of the concepts do also appear in physics books, and in books on Lie groups and representation theory for Lie groups. There are some books on quantum field theory and particle physics which use Lie algebras with little or no reference to the Lie groups from which they are constructed.

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