# Prerequisites for Lie groups and Lie algebras

I'm thinking of auditing the Introduction to Lie groups and Lie algebras course next year at my university.

Will Real Analysis at Bartle or Rudin or Tao level, Linear Algebra at Kunze level and Group, Rings and Modules at Dummit and Foote level be okay to take this course?

Will I require anything else??

• What/how much multivariable calculus have you done? May 13, 2021 at 4:15
• You'll most likely be fine with that background. Of course, as Elliot mentioned, multivariate calculus is a good background to have as well May 13, 2021 at 4:18
• Marsden, Tromba Vector Calculus level May 13, 2021 at 4:18
• We didn't do Differential forms, Inverse and Implicit function theorem though. Are they required May 13, 2021 at 4:19
• Well they are both covered in Marsden/Tromba, so they shouldn't be too difficult to learn. May 13, 2021 at 4:20

It depends. A rigorous treatment of Lie groups and Lie algebras (as in the style of, say, the excellent book Lie Groups: Beyond an Introduction by Knapp) does require a solid background in differential geometry and manifold theory. If the class is taught in this way, you may need to wait a year or two until after you have developed some expertise in these topics.

However, it is possible (and commonplace, these days) to provide an introduction to the subject 'via matrix groups' (as in the style of, say, Matrix groups for undergraduates by Tapp or Lie groups, Lie algebras and Representations by Brian C. Hall), where the topic is approached by beginning with the matrix groups we all know and love ($$\operatorname{GL}_n, \operatorname{O}_n, \operatorname{SO}_n, \operatorname{U}_n, \operatorname{SU}_n$$, etc.) and covering a lot of the theory using only these elementary groups. If the class is taught this way, you should be fine with the background you have.

At the end of the day, the best course of action is probably just to email the lecturer, and ask what approach they are taking, and what prerequisites they expect of their students.

• Yeah, our professor will take the latter approach and is probably going to follow Brian C. Hall's book itself. May 13, 2021 at 4:38
• Okay, well Hall only assumes multivariable calculus and linear algebra, so you're set! And actually, now you mention it, some general topology is handy. As for rep. theory, he uses some rep. theory results, but they are introduced 'along the way'. Good luck! May 13, 2021 at 4:40
• Will I require a whole course in General Topology, or concepts like compactness, connectedness, metric spaces be enough? May 13, 2021 at 4:49
• If the class is based on Hall's book, they should be enough. But remember, you will never be 100 percent prepared. The key thing is that you are interested/hardworking enough to persevere. May 13, 2021 at 4:54

If your professor plans to go through the route of the standard matrix groups which is no doubt an excellent way to learn Lie algebras then honestly, you don't need more prerequisites than Linear Algebra. You need not be too worried however, just keep in mind that you might have to learn a few basic topological things on the fly like connectedness, compactness etc. if you haven't already.

There might be some technical theorems and material which you might not have the background to immediately understand. As another commenter mentioned, if the professor follows Hall's book, you'll be fine. In my experience, I started learning Lie groups from this book by myself at a time when I had very limited background.

If you think you can handle a more comprehensive and rigorous treatment of the subject, check out Fulton & Harris, Humphreys or the Berkeley notes.