Nice pedagogical explanation! This topic really deserves to be seen from as many different directions as possible. This is not my first introduction to Lie algebras but some pins dropped during this lecture.
In the beginning half hour or so I was intrigued by all these pretty diagrams and actually learned something. At about 44 minutes you lost me too. Did anyone survive till the end? Now I just have a lot of pretty diagrams with some half understood meaning. The point of all this remains in the dark even though I kind of understood certain Lie algebras from the start. How to get to the next level?
I suppose the next level is to work on actually understanding Lie algebra modules, how the E_i:s, F_i:s and H_i:s act on them in details. Also, it is important to understand how the [E_i, E_j]:s and [F_i,F_j]:s etc work within the Lie Algebra, and how they act on its modules, although i think this is pretty easy, using repeated Lie bracketting/action. Of course we need an understanding of which shapes of modules are allowed, especially when it comes to Lie algebras that have Dynkin diagrams that are not simply laced. It is important to take notes on the weights of the "roots" of a module, so we can find out which one has the highest weight (relative to the chosen simple positive root E_i:s of the Lie algebra).
I actually kind of get it now! At least in theory, still need to practice some examples. The only really tricky thing is to correctly differentiate between the various H_i:s of a rank >1 Lie algebra, remembering how the "parallellograms" do not "commute", and especially how to lift this to the roots of the modules (getting different "repeated" module roots in the "same place"). Btw i think the example picture of the module of sl(3) is somewhat wrong, because it actually contains "commuting parallellograms", which should not be present. If i understand correctly.
that picture describes a 5 dimensional representation of sl(2). Hence those dots are any basis and the arrows describe the action of each generator of sl(2).
This is the most straightforward explanation of Dynkin diagrams I could find on CZcams.
I agree with you! It is a good lecture!
Sick! I love this knowledge being so open on the internet.
Nice pedagogical explanation! This topic really deserves to be seen from as many different directions as possible. This is not my first introduction to Lie algebras but some pins dropped during this lecture.
In the beginning half hour or so I was intrigued by all these pretty diagrams and actually learned something. At about 44 minutes you lost me too. Did anyone survive till the end? Now I just have a lot of pretty diagrams with some half understood meaning. The point of all this remains in the dark even though I kind of understood certain Lie algebras from the start. How to get to the next level?
I suppose the next level is to work on actually understanding Lie algebra modules, how the E_i:s, F_i:s and H_i:s act on them in details.
Also, it is important to understand how the [E_i, E_j]:s and [F_i,F_j]:s etc work within the Lie Algebra, and how they act on its modules, although i think this is pretty easy, using repeated Lie bracketting/action.
Of course we need an understanding of which shapes of modules are allowed, especially when it comes to Lie algebras that have Dynkin diagrams that are not simply laced.
It is important to take notes on the weights of the "roots" of a module, so we can find out which one has the highest weight (relative to the chosen simple positive root E_i:s of the Lie algebra).
Very interesting.i liked it
I actually kind of get it now! At least in theory, still need to practice some examples.
The only really tricky thing is to correctly differentiate between the various H_i:s of a rank >1 Lie algebra, remembering how the "parallellograms" do not "commute", and especially how to lift this to the roots of the modules (getting different "repeated" module roots in the "same place").
Btw i think the example picture of the module of sl(3) is somewhat wrong, because it actually contains "commuting parallellograms", which should not be present. If i understand correctly.
..the diagrams explained..at last......!!!!!!!!!!
I didn't get why you took 5 basis elements in 03:59 since sl(2) is 3-dimensional. What are these five basis elements?
that picture describes a 5 dimensional representation of sl(2). Hence those dots are any basis and the arrows describe the action of each generator of sl(2).
interesting!
Ничего не понял, но очень интересно
I m here after an year
Learned basic Abstract algebra
So that I could understand it
But still nothing........😓
😔I think, am useless
Don't feel that bad, Lie Algebras are a whole can of worms on their own.