Using Michael Artin's "Algebra" for Group Theory

[WWCY]

Hi all,

I have stumbled upon Artin's book "Algebra" and was wondering if I could use it to do some self-study on Group Theory.

Some background: I am a physics undergraduate who has some competence in elementary logic, proofs and linear algebra. It seemed to me that ideas related to Group Theory were getting more prevalent in higher-level QM courses and I decided that I'd have to read up a little on the side.

I managed to obtain this book for free but was unsure what to do with it, hence this question.

Of course, other suggestions regarding "starter" texts are welcome.

Thanks in advance!

 

[smodak]

1. Physics from Symmetry by Schwichtenberg
2. An Introduction to Tensors and Group Theory for Physicists by Jeevanjee
3. Group Theory in a Nutshell for Physicists by Zee
4. Quantum Theory, Groups and Representations: An Introduction by Woit (not as an introduction :) but a great book nonetheless)

 

[MidgetDwarf]

Not sure if the physics side. But it is a rigorous math book.

So I'm not sure how applicable it may be for physicist. For a mathematics major it a perfect fit. Author is very good at explaining ideas, and ties everything nicely.

What book was you're linear algebra class based on?

And by proofs in linear algebra. Did you prove stuff like the fundamental theorem of linear maps, "plus or minus theorem", properties relating to linear transformations etc??

 

[FourEyedRaven]

Hi

The book is a standard reference in abstract algebra, and Michael Artin was a professor at MIT, so it's not a bad quality book.

If you read the first half of the book, up to group representations, you can learn the standard material of groups theory, review/learn linear algebra, understand the connection between groups and symmetric transformations, learn about linear groups, and learn some representation theory.

Given how the author presents group theory interwoven with linear algebra, it will be good for physics students looking for a solid mathematical preparation for quantum mechanics. My sense is that, once you learn this material, you can hit the ground running on just about any further topic in group theory required for quantum physics.

The only possible exception is when group theory meets differential geometry or topology, because these subjects require learning in some depth before being able to combine with groups. But I suspect this only comes up very late in a physics degree, at a MSc. or PhD level.

 

[WWCY]

1. Physics from Symmetry by Schwichtenberg
2. An Introduction to Tensors and Group Theory for Physicists by Jeevanjee
3. Group Theory in a Nutshell for Physicists by Zee
4. Quantum Theory, Groups and Representations: An Introduction by Woit (not as an introduction :) but a great book nonetheless)

Many thanks, I'll give these books a look!

 

[WWCY]

Thanks for your response!

Not sure if the physics side. But it is a rigorous math book.

So I'm not sure how applicable it may be for physicist. For a mathematics major it a perfect fit. Author is very good at explaining ideas, and ties everything nicely.

What book was you're linear algebra class based on?

And by proofs in linear algebra. Did you prove stuff like the fundamental theorem of linear maps, "plus or minus theorem", properties relating to linear transformations etc??

Regarding your first question, it was based on "Elementary Linear Algebra" by Anton Rorres, though the actual material covered was far more watered down, which leads on the your second question.

Unfortunately I have not learned any of what you mentioned, we did stuff up to invertible matrices and simple transformations and that's it (before transitioning to multi-variable calculus). I take it that the concepts you brought up are necessary for the understanding of Artin's book? If so I'd better start looking at some of the other suggestions!

 

[FourEyedRaven]

Thanks for your response!
Regarding your first question, it was based on "Elementary Linear Algebra" by Anton Rorres, though the actual material covered was far more watered down, which leads on the your second question.

Unfortunately I have not learned any of what you mentioned, we did stuff up to invertible matrices and simple transformations and that's it (before transitioning to multi-variable calculus). I take it that the concepts you brought up are necessary for the understanding of Artin's book? If so I'd better start looking at some of the other suggestions!

You don't have to give up on Artin's book. On the contrary! I thought you had learned more advanced linear algebra in college, instead of the material necessary to understand multivariable calculus. This book will introduce you to the kind of linear algebra that is used in the formalism of quantum mechanics. And the first chapter is a review of the material you most likely learned for multivariable calculus. It actually looks like a perfect fit for your goals.

 

[MidgetDwarf]

Thanks for your response!
Regarding your first question, it was based on "Elementary Linear Algebra" by Anton Rorres, though the actual material covered was far more watered down, which leads on the your second question.

Unfortunately I have not learned any of what you mentioned, we did stuff up to invertible matrices and simple transformations and that's it (before transitioning to multi-variable calculus). I take it that the concepts you brought up are necessary for the understanding of Artin's book? If so I'd better start looking at some of the other suggestions!

I am familiar with Anton :). I also used it + Lang in my intro linear class. The reason why I asked was that "quality" math books require something called mathematical maturity. This is gained by reading math textbooks, doing proofs, and problems. The only prerequisites are that you know how proofs work. But you said you know logic and proofs, so you have this down. You do not need the topics I mentioned in my first post to work through Artin. But it makes the first, third, and 4th chapter easier to digest. First chapter deals with matrix multiplication, facts about matrix multiplication, and determinants. The third chapter deals with Vector Spaces and facts related to them, and the 4th (if I remember correctly), deals with things like basis, null space etc. This helps students understand the General Linear Group, which Artin emphasizes over the Symmetric Group (most authors emphasize the Symmetric Group). So these chapters, will teach you some upper division Linear Algebra, which FourEyedRaven commented on.

The book is well written. But you have to be careful when reading. Ie., A law of composition on a set. The definition of a law of composition uses the word function. But if a person does not know what a function is, they would never understand well defined/ closure is included in the definition of a law of composition on a set. Also, you have to get you're hands dirty and verify some of the results in the passage. But, the book is well written so this is not so hard. It lacks examples compared to other text.

I would say purchase the book, and read it in conjunction with another book on Algebra. Maybe read it together with Pinter: A book on Abstract Algebra. I like Pinter. Fraileigh? But I do not like Fraileigh for later chapters, it has good problems however. If you find Artin difficult, you can always return to it later. I remember purchasing Axler Linear Algebra Done Right, and It was above my level at that time. But I returned to it 1 year later, and was able to finally work through it, complete most of the problems, and understand it.I would maybe send a message to Micromass or Mathwonk in regards to using Artin. Mathwonk recommended Artin to me, but he is Mathematician. Maybe he dabbles in physics but I'm not sure. Micromass struck me as mathematical physicist, but again i am not sure.

 

[mathwonk]

I am indeed a mathematician, but agree with raven that artin is a good choice for physicists precisely because he does groups in connection with linear algebra, i.e. he treats both finite and infinite groups, as a physicist needs. In fact artin is almost the only intro to algebra book i know of that treats linear groups. the only caveat with artin is that although it is an intro, it is still somewhat terse compared to more elementary works out there. So you have to read it more slowly. but it is excellent.

 

[TeethWhitener]

Depending on what physics you're looking for, "Group Theory and Quantum Mechanics" by Michael Tinkham might merit a look. It's geared less toward particle physics and more toward atomic/molecular/solid state. Crystal symmetries and things of that nature.