# Lie groups, Lie algebra books?

I'm looking for a solid book on Lie groups and Lie algebras, there is too many choices out there. What is a classic text, if there is one?

Related Science and Math Textbooks News on Phys.org
George Jones
Staff Emeritus
Gold Member
I'm looking for a solid book on Lie groups and Lie algebras, there is too many choices out there. What is a classic text, if there is one?
Tough question.

How much rigor?

Are you interested because of the beautiful math?

Because of applications to elementary particles and quantum field theory?

Just Lie groups and Lies algebras, or representations, too.

Just pure maths, grad or undergrad, but rigor not blown out of proportions. My end goal is to understand E8, and see what's it all about. Some texts I looked at are aimed at physicists, like Gilmore, or Lipkin. Amazon generates a continuous spectrum of those, I need to settle on one book (eigen-book if you will), with a reasonable eigen-price.

I would also like to learn about Lie Algebra for the same reason as waht. E8 seems very interesting and Lie Algebra just seems so fundamental for quantum field theory. A great tool to have, but how should one learn it? Waht, have you checked out Lisi's paper, "An Exceptionally Simple Theory of Everything?" I am having trouble understanding much of anything from it, and I think learning Lie Algebra would be a great start.

Here's his paper: http://arxiv.org/abs/0711.0770

I too am having trouble with Lie algebras - I'm just getting into classical field theory, never mind QFT. I'm going to need to brush up on my maths to get anywhere with this.

The main problem that's foxing me with them at the moment, and I'd like to find a good answer for, is why it's the commutation relations that are fundamental and not the operators themselves.

George Jones
Staff Emeritus
Gold Member
Just pure maths, grad or undergrad, but rigor not blown out of proportions. My end goal is to understand E8, and see what's it all about.
"[URL [Broken]
Semi-Simple Lie Algebras and their Representations[/URL] by Robert Cahn is a free book (wasn't free when I picked it up!) on Lie algebras that has a chapter on the exceptional algebras. This book was written for physicists, but doesn't refer to any physics applications.
"[URL [Broken]
Representation Theory: A First Course[/URL] by Fulton and Harris, a standard math text, starts off terse, but becomes very readable and (maybe too) expansive in its middle. It's cool to see quark multiplet diagrams (as representations of $sl\left(3 , \mathbb{C}\right) \cong \mathbb{C} \otimes su\left(3\right)$) appearing in a pure math book, even though the book doesn't identify them as such.

I have never really looked at the representations of the exceptional Lie algebras, so I can't comment on the treatment in either book.

Last edited by a moderator:
I would also like to learn about Lie Algebra for the same reason as waht. E8 seems very interesting and Lie Algebra just seems so fundamental for quantum field theory. A great tool to have, but how should one learn it? Waht, have you checked out Lisi's paper, "An Exceptionally Simple Theory of Everything?" I am having trouble understanding much of anything from it, and I think learning Lie Algebra would be a great start.
Yes, I'm familiar with this paper, it's the main reason why I want to learn all the Lie stuff.

"[URL [Broken]
Semi-Simple Lie Algebras and their Representations[/URL] by Robert Cahn is a free book (wasn't free when I picked it up!) on Lie algebras that has a chapter on the exceptional algebras. This book was written for physicists, but doesn't refer to any physics applications.
Thanks George, that's an excellent book on Lie Algebra, already learned something at first glance.

Last edited by a moderator:
I am actually having trouble viewing this book. What do I need in order to view it?

CompuChip