Do I need a lot of abstract algebra knowledge to start learning Lie algebra

[kof9595995]

I'm a physics undergrad and doing some undergrad study on QFT, and I found that Lie algebra is often invoked in texts, so I decide to take a Lie algebra this sem but I've not taken any abstract algebra course before.The first day's class really beats me because the lecturer used many concepts from abstract algebra, as I anticipated. I just want to ask do I need a full knowledge of abstract algebra to learn Lie Algebra, or just some basic concepts?

 

[kof9595995]

Sorry, I intended to put this in "Academic guidance", clicked wrongly...

 

[twofish-quant]

I anticipated. I just want to ask do I need a full knowledge of abstract algebra to learn Lie Algebra, or just some basic concepts?

I don't think that you need much abstract algebra to learn Lie Algebra. I found that you need to understand differential equations pretty well for Lie Algebra to make much sense.

 

[kof9595995]

Thank you, then I guess I'll stick to the class and try to catch up with some abstract algebra by myself.

 

[George Jones]

I don't think that you need much abstract algebra to learn Lie Algebra. I found that you need to understand differential equations pretty well for Lie Algebra to make much sense.

I suppose it depends on the style of presentation. I took graduate courses in Lie algebras and representation theory from the math department. My previous exposure to abstract algebra was very useful, and nary a differential equation in sight.

 

[mathwonk]

This is one subject I never learned, but I have heard a few things. Given a lie group, which is in particular a differentiable manifold, the associated lie algebra is the tangent space at the identity element. There is a "bracket" multiplication involving vector fields that defines an algebra structure on that tangent space, making it an abstract "lie algebra". The study of vector fields is of course the geometric version of differential equations.

However one can also simply study algebras with that type of multiplication, calling them "lie algebras", without reference to manifolds, vector fields, and hence differential equations. Indeed I have met people who thought they knew about lie algebras who did not even know the connection with lie groups, having read only abstract treatments in purely algebraically oriented books.

There seems to be a decent introduction to lie algebras on wikipedia.

 

[axiomatic7777]

From what I noticed from sitting in class: specific concepts from abstract algebra is not extensively used, but you should at least know the basics (i.e. what is a group, subgroup, field, some knowledge about matrix groups, etc.). After all lie groups are still groups :)

Depending on the level of your course (undergrad or grad?) and your previous experience in higher level proof-based math, you might have difficulties jumping right into lie groups since it is probably assumed that you know a lots of the techniques in writing and understanding proofs (it does not come easily except with lots of practice, trust me! I personally found abstract algebra the hardest math class to get started on since the way of thinking in algebra is so different from whatever I learned before).

I would also recommend reading some kind of "lie group for physicists" type of book as a supplement since it will give you some concrete physics examples and/or motivations for studying some seeming abstract concepts.

 

[mathwonk]

Does your background include calculus on manifolds, and differential geometry? Lie algebras are an abstraction of the study of vector fields on manifolds, and in fact all occur as the lie algebra associated to a manifold which is a group, i.e. a lie group.

The lie bracket multiplication on vector fields measures whether or not those vector fields can be presented as the standard unit vector fields in some coordinate system. For lie groups, one considers only invariant vector fields for the group law, and then the answer is that (independent) invariant vector fields can always be so represented if and only if the group is abelian.

A connected abelian lie groups seems to be a product of a compact torus and some R^n.

Since lie algebras seem to be an algebraic tool for studying differential manifolds, it seems hard to learn the theory without those prerequisites. Of course some people teach things "in a vacuum" without explaining connections to other topics, even ones that gave rise to them..

 

[mathwonk]

Have you encountered the frenet - serret formulas in elementary differential geometry? As I recall, these are a reflection of the lie algebra structure of the "orthogonal" group of rotations of 3- space. Maybe you really only need to know the classical "linear" lie groups and their lie algebras, i.e. matrix groups. This subject is much more concrete, and essentially a branch of linear algebra.

 

[twofish-quant]

I suppose it depends on the style of presentation. I took graduate courses in Lie algebras and representation theory from the math department. My previous exposure to abstract algebra was very useful, and nary a differential equation in sight.

My first exposure to Lie Algebras was in the physics department in the context of advanced classical and quantum mechanics. They teach you mechanics, then they point out that are these things called generators, and then they point out that you can do algebra with them with these things called commutators. It's only after they they go through the physics that they mention "oh, by the way, if you hear a mathematician talk about a Lie Algebra, that's what they are talking about."

Also the fact that there are very different ways of explaining what a Lie Algebra is, is what makes them very useful.