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Naive Lie Theory
In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-called "classical groups'' that capture the symmetries of real, complex, and quaternion spaces. These symmetry groups may be represented by matrices, which allows them to be studied by elementary methods from calculus and linear algebra. This naive approach to Lie theory is originally due to von Neumann, and it is now possible to streamline it by using standard results of undergraduate mathematics. To compensate for the limitations of the naive approach, end of chapter discussions introduce important results beyond those proved in the book, as part of an informal sketch of Lie theory and its history. John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer, including The Four Pillars of Geometry (2005), Elements of Number Theory (2003), Mathematics and Its History (Second Edition, 2002), Numbers and Geometry (1998) and Elements of Algebra (1994).
218 pages, Hardcover
First published January 1, 2008
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Displaying 1 - 8 of 8 reviews
July 17, 2014
This was a very short, but still very informative, book about matrix Lie groups. It discusses the classical matrix groups (general linear, special linear, orthogonal, unitary, and symplectic) and their relation to their corresponding Lie algebras. The correspondence is given by the matrix exponential and logarithm functions, and this allows us to see which Lie algebras and which Lie groups are simple, with the help of some analysis and topology.
This book definitely assumes knowledge of basic group theory, linear algebra, and analysis, but it also gave very nice reviews of these topics which made the text much easier to follow. Overall, I found this a very cute introduction to certain types of groups that I haven't studied in depth in my previous Algebra studies.
This book definitely assumes knowledge of basic group theory, linear algebra, and analysis, but it also gave very nice reviews of these topics which made the text much easier to follow. Overall, I found this a very cute introduction to certain types of groups that I haven't studied in depth in my previous Algebra studies.
August 31, 2019
If you’re already on the fence about buying this book, get it. And Stillwell’s other books.
If you’ve heard of quaternions via graphics or robot control, consider getting this after reading David Lyons� essay on quaternions and watching Niles Johnson’s Hopf fibration video.
It's probably the best place to start if you work with matrices but don't know about Lie theory.
Lie theory (orthogonal, unitary, and spin groups) can give you a geometric view into rotations, which you might know are implicit in matrices but not have thought through clearly.
Lie theory also connects matrices, complex numbers, and quaternions. (Quaternions are rotations in 3-space. Read Baez.)
If you’ve heard of quaternions via graphics or robot control, consider getting this after reading David Lyons� essay on quaternions and watching Niles Johnson’s Hopf fibration video.
It's probably the best place to start if you work with matrices but don't know about Lie theory.
Lie theory (orthogonal, unitary, and spin groups) can give you a geometric view into rotations, which you might know are implicit in matrices but not have thought through clearly.
Lie theory also connects matrices, complex numbers, and quaternions. (Quaternions are rotations in 3-space. Read Baez.)
April 22, 2024
A really solid introduction to Lie Theory that uses only undergraduate level mathematics.
Lie theory is an interesting intersection of two families of ideas: groups and topology. What happens if a group is also a manifold?
This introduction was clear, comprehensive enough for me to get what I wanted out of it, and did not require loads and loads of background knowledge. It deals entirely with matrix Lie groups and requires no pre-existing topology. I shopped around for the right Lie Theory text for a long time before finding this one, and I'm glad I persevered; there are many other texts and this one really fit my requirements (or perhaps the others were just bad?). I also really appreciated how he motivated the next mathematical step, showing how it links into the bigger picture and what Lie theory is being designed to do.
Occasionally I was stumped by a proof, either by confusion (Eichler's proof of the Baker-Campbell-Hausdorf theorem) or by boredom (the long tedious proofs of the simplicity of certain groups). At some point I should clear up the confusion (I will be leaving the boredom alone though).
But overall, a great book.
Lie theory is an interesting intersection of two families of ideas: groups and topology. What happens if a group is also a manifold?
This introduction was clear, comprehensive enough for me to get what I wanted out of it, and did not require loads and loads of background knowledge. It deals entirely with matrix Lie groups and requires no pre-existing topology. I shopped around for the right Lie Theory text for a long time before finding this one, and I'm glad I persevered; there are many other texts and this one really fit my requirements (or perhaps the others were just bad?). I also really appreciated how he motivated the next mathematical step, showing how it links into the bigger picture and what Lie theory is being designed to do.
Occasionally I was stumped by a proof, either by confusion (Eichler's proof of the Baker-Campbell-Hausdorf theorem) or by boredom (the long tedious proofs of the simplicity of certain groups). At some point I should clear up the confusion (I will be leaving the boredom alone though).
But overall, a great book.
August 23, 2024
A great way to lure non-mathematicians into the world of topology and manifolds, using mathematics that is familiar to most engineers.
Came for the quaternions, stayed for the elementary proofs within group theory. Now left because the book doesn’t go into great detail about topology (which is arguably a big thing for Lie Groups).
Came for the quaternions, stayed for the elementary proofs within group theory. Now left because the book doesn’t go into great detail about topology (which is arguably a big thing for Lie Groups).
June 2, 2021
As always this is another great book by John Stilwell and it is one of the best elementary introductions to the Lie Theory.
July 19, 2022
An exceptionally clear introduction to Lie theory.
August 15, 2015
(warning: this review only concerns first 4 chapters) nice intro to Lie Theory. I'm not a mathematician. I started reading this book because I came across relevant concepts in a few mathematical biology papers, and wanted to know more about what they are talking about. Quaternions definitely got me pretty excited! The book enticed me to read on and on, more than I planned to and have time for. Finally I quit at chapter four. The reason is that there is a lack of examples and more concrete exercises for self-learning, and some definitions/theorems/proofs are not presented as technical as to be very clear. I imagine that things would be much smoother if there's an instructor to guide one through this book. But for me, I felt that my grasp of the first four chapters is not unshakable enough to venture beyond.
Overall it's a great book and I may come back to it and revise this review.
Overall it's a great book and I may come back to it and revise this review.
March 23, 2013
my 5 stars is contingent on the book accompanying a corresponding lecture series, as I think it is intended to do. while no individual equation is out of reach for the undergrad, some of the conceptual leaps would I think be too far for those not already versed in quaternions and group theory. that said I would take a class using this book in a heartbeat.
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