Suggest a text for Lie Groups, Lie Algebras, Differential Geometry, etc. (not Frankel)

  • #1
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Main Question or Discussion Point

Summary:: Suggest a textbook

Good Morning

I have repeatedly tried to read Frankel's "Geometry of Physics" and I get swamped and overwhelmed.

(I hasten to add that as a MECHANICAL engineer, my math background has been deficient.)

I retire in about 10 years and I am looking to learn the material in his book, for the fun of it -- just the fun of it.

However, the book is dense, and assumes great facility with "ALREADY KNOWING" what he is talking about.

I am more interested in the overall picture to understand issues like: Lie Groups, Lie Algebras, Differential Geometry, Tangent Bundles
Co-tangent, co-vectors, dual spaces, etc.

I do understand the concepts (generally) behind these terms, but they do not come to me with ease.

I would prefer one text book.


Are there any suggestions?
 
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Answers and Replies

  • #2
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I would suggest you to read my insight articles here (and / or the references there), e.g.
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-basics/
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/
but they are of course not on textbook level. I doubt you will find one book that covers it all. Moreover there is a discrepancy between the mathematical and the physical treatment: in language as well as in presentation. The articles I cited and their references are from a mathematical point of view, so they might not be what you were looking for.
 
  • #3
phyzguy
Science Advisor
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Personally, I like "Geometry, Topolgy, and Physics", by Mikio Nakahara.
 
  • #4
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I would suggest you to read my insight articles here (and / or the references there), e.g.
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-basics/
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/
but they are of course not on textbook level. I doubt you will find one book that covers it all. Moreover there is a discrepancy between the mathematical and the physical treatment: in language as well as in presentation. The articles I cited and their references are from a mathematical point of view, so they might not be what you were looking for.

Most of all: THANK YOU for responding.

But please forgive me, (this is NOT a commentary on your work), but I am looking for a book that would help me understand what you did here. (Yes, I have seen this before).

I am looking for a little less mathematical rigor (to compensate for my poor mathematical training), and a bit more physics and discussion.

I do hope to one day understand what you have done, but I am not there yet.
 
  • #5
troglodyte
Gold Member
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Most of all: THANK YOU for responding.

But please forgive me, (this is NOT a commentary on your work), but I am looking for a book that would help me understand what you did here. (Yes, I have seen this before).

I am looking for a little less mathematical rigor (to compensate for my poor mathematical training), and a bit more physics and discussion.

I do hope to one day understand what you have done, but I am not there yet.
You can try gauge fields ,knots and gravity,John Baez
 
  • #6
mathwonk
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Homework Helper
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In reply to your first post, I suggest you look at "Foundations of differentiable manifolds and lie groups", by Frank W. Warner. Of course in a later post you say you are not so interested in mathematical rigor, and this book is quite mathematically rigorous. But my belief is that the fact it covers the mathematics you want to know, without assuming most of it in advance, means it could help you. E.g it states and uses the inverse function and implicit function theorems, which are absolutely basic to understanding manifolds. In my experience most graduate students are not at ease with them, even though they are fundamental to every advanced calculus course, or should be. The book is thus at the level of a beginning graduate course, say in mathematics.

It is quite efficient and goes quite far, covering briefly at least all the topics in post #1, and also proves advanced theorems that are not at all easy to find at such an accessible level, such as the Hodge theorem. It does assume you know the definition of the dual space V* of a vector space V. I.e. it assumes you do have some finite dimensional linear algebra experience. In this case of course, V* = the space of all linear functions f:V-->F, where F is the field of scalars of V. If v1,...,vn is a basis of V, then the functions f1,...,fn, where fj(vj) = 1, and fj(vi) = 0 for i≠j, is a basis for V*. Hence in the case of finite dimensions, V and V* have the same dimension, hence are isomorphic. But there is no natural isomorphism between them until you first choose a basis for V, (or for V*).

Since it covers all your topics it moves quite rapidly, so in reading it you should move quite slowly.

Another suggestion, if you want to learn about lie brackets in a motivated way, is to read a differential geometry book where they arise somewhat naturally as operations on vector fields.

First of all, I declare (as will become obvious to some) I myself know almost nothing about this subject, but still making bold to give suggestions, subject to correction by experts here, I make a few remarks. In chapter 5, vol. 1, of Spivak's differential geometry, he makes crystal clear that when a vector field is viewed as a differential operator on functions and forms, one can define a sort of multiplication of vector fields by simple composition, differentiating repeatedly. Then it turns out that the Lie derivative of one vector field wrt another, coincides with the vector field obtained by "commutation" of this operation, i.e. it measures the difference in composing differential operators in one order, and subtracting the result in the other order. This called the bracket.

Now what is the importance of this operation? If we recall the wonderful rule from calculus of "equality of mixed partials", it turns out this equality is exactly the statement that the bracket is zero for differentiation wrt different coordinate variables. The wonderful fact is that this phenomenon is precisely what allows you to recognize when a family of n vector fields on an n - manifold, does in fact arise from the n variables of a local coordinate system.

Now, I am getting even further from my expertise here, but in case the manifold is also a (smooth, i.e. lie) group, the group operation allows you to take just one tangent vector at the origin, and translate it around to form a vector field on the whole manifold. In this way we can define a bracket multiplication just on the tangent space at the origin, making it the lie algebra associated to that lie group. So this algebra "linearizes" the group in the same way that a tangent space linearizes a manifold, at least locally. Further, there is a nice smooth map, the "exponential map", from the tangent space at the origin to the manifold, allowing you to relate the group and the algebra. In the case of the group of invertible nxn matrices, the algebra is the vector space of all nxn matrices, and miraculously, the exponential map is even given locally, by the usual exponential series applied to a matrix!

Ok, I'm outta here. Hope I have not done too much harm. Remember these are the views of someone who does not know the material, and has only perused the first few pages of some books. Enjoy reading some experts!

added later: Wow, hoping to learn something about why physicists care about this, I perused the wikipedia article on particle physics and representation theory. It seems one needs to know an awful lot more math for all this, e.g. covering spaces and fundamental groups, extensions of groups and cohomology of groups, as well as hilbert spaces. For encouragement in this regard, I always liked the apocryphal quote from Hilbert: "so just what is this Hilbert space?"
 
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