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- Thread starter kof9595995
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Sorry, I intended to put this in "Academic guidance", clicked wrongly...

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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.

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George Jones

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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.

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mathwonk

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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.

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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.

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mathwonk

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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..

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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.

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