When he pointed out that the points at the origin of the root map corresponded to the 2-dimensional Cartan subalgebra, everything clicked. This _really helps_ with understanding particle physics. (btw, the algebra is su(3), in case people didn't realize)
@thiagolucena4366
2 жыл бұрын
This comment deserves more likes!
@EivindDahl
7 жыл бұрын
This guy is among the most organised lecturers I have ever seen
@kevinleeraha6486
7 жыл бұрын
Eivind Dahl One of the most knowledgable too. He seems to know every discipline of mathematics as long as they have anything to do with physics (differential geometry, functional analysis, measure theory, etc.).
@tataltsal
Жыл бұрын
A geometric meaning of the Weyl transformation is a simple reflection. I think mentioning this would have been awesome.
@millerfour2071
3 жыл бұрын
6:09, 15:20, 36:54, 39:50, 48:14, 1:02:45 (no need to explicitly calculate dual killing form), 1:11:26, 1:14:27 (take all possible “horizontal reflections”), 1:31:02, 1:36:36, 1:37:40
@somasundaramsankaranarayan4592
7 жыл бұрын
At 47:47, it was wrongly assumed that the three vectors are linearly independent, when indeed they are linearly dependent.
@magerfam
7 жыл бұрын
i agree. for example alpha=gamma and beta = -gamma....
@mikeeeyaleeex2615
7 жыл бұрын
is there any book of lie groups and it's lie algebra that uses this method to find the lie algebra of the lie group ? because i tried to hard to find the lie algebra of SO(3) using this method but i failed too many because of the issue of parameterization of SO(3) .thanks.
@kevinleeraha6486
7 жыл бұрын
magerfam you can just check by computing the det and it is 0.
@kevinleeraha6486
7 жыл бұрын
The three vectors on the rhs aren't linearly independent, but the general vector and two other vectors are linearly independent unless a = 0 (can check by computing the associated determinant).
@vadimostapenko5063
8 жыл бұрын
Very nice and tasty way to present those computations. Thank ytou a lot!
@rodrigomendes8
2 жыл бұрын
I made the calculations for the killing form and found k22=0. Don't know where I'm wrong. The calculations: K22=C^{m}_{2 n}C^{n}_{2 m} =C^{1}_{2 n}C^{n}_{2 1} + C^{2}_{2 n}C^{n}_{2 2} + C^{3}_{2 n} For the first term, n=3 but C^{3}_{2 1} = 0. The second possibility is on the second term n=2, but C^{1}_{2 2} = 0. So, the first two term goes to zero. The second term is zero and the third is zero to a think similar to the first that I made. So, k22= 0
@reinerwilhelms-tricarico344
4 жыл бұрын
The captions displayed the Weyl group and transformations as wild group/transformations - consistently. 😂
@borispawilowski23
4 жыл бұрын
At 01:16:00 one can actually find s_\pi_1 (\pi_1+\pi_2) by linearity by using the previous calculus
@peterhunt1968
3 жыл бұрын
Awesome 👏
@pettPette
8 жыл бұрын
I like all your lectures
@varunmenon830
4 жыл бұрын
At 1:25:28 why are the roots evaluated at the basis vectors of H the same numbers on the root diagram? Is the root diagram necessarily w.r.t the h1 h2 basis?
@ernestomamedaliev4253
4 жыл бұрын
As far as I understood, we are looking for a basis of the Lie algebra and its associated structure constants. Therefore, we are looking for a basis of the Cartan subalgebra, namely h1 and h2 in this case. From the Dynkin diagram, we are dealing with a basis of the dual space of H. Therefore, once we are doing the representation in coordinates, we are choosing the h1 and h2 mentioned in the previous paragraph. The coordinates of a vector are given w.r.t. some basis. This is why he freely says that pi1=(1,0). We could say pi1=(1/sqrt(2), 1/sqrt(2)) and it would have associated another basis [the overall result would be just a rotation of the root representation by 45º]. So, we choose h1 and h2 in order to have that representation of the roots. In any case, it is clear that the components of a vector are given always w.r.t. some dual basis. That is the way of defining the components of a vector. Coordinates of a vectos has no meaning if a basis is not given.
@brandonwillnecker8060
2 жыл бұрын
You can imagine that the roots, the lambdas, are covectors because they are from H*. The vectors are then the h1 and h2 from H. How do we get the components of a covector given a basis for the vectors? The components of the covector are found by acting the covector on each basis vector. So here the first component of a given lambda is given via lambda(h1) and the second component is given via lambda (h2) where h1 and h2 are the basis vectors of H.
@abomohamed7919
6 жыл бұрын
I want to know is that there is a computer program by formal calculation that allows to build the dynkin diagram. thanks
@joelcurtis7447
6 жыл бұрын
When he draws the picture of the fundamental roots around 1:14:00, what is the space we're in? What is represented by each axis? Are they just the real numbers, since this is the realification of the dual space of the cartan subalgebra H?
@neelmodi5791
6 жыл бұрын
The fundamental roots can be realized as a basis of R^2, so he's just drawing what they'd look like if you drew them in R^2. And yeah, the axes are simply real axes. It doesn't matter how you draw the axes though, but for convenience (to make the coordinates nicer) he draws the first axis along the same direction as the first fundamental root.
@luisgeniole369
4 жыл бұрын
Did you know that german professors work part time as window cleaners?
@andreemcaldas
6 жыл бұрын
At 50:36. After you know that alpha is zero, you are back to the first, already analyzed case.
@shanborlangbynnud2320
4 жыл бұрын
I don't know what magic he's excecuting at 1:12:57 can anyone help.
@ernestomamedaliev4253
4 жыл бұрын
He is just calculating where the point lands at 120 degrees in the unit circle. Coordinates are (cos(120º), sin(120º))
@gubazoltan99
2 жыл бұрын
He applies to Weyl group to the fundamental roots to recover the roots space. He fixes pi1 to lie on the horizontal axis and then finds to other roots with respect to pi1. He uses the fact that the killing form K* defines an inner product in the root space so one can find the "angles" between the roots.
@jamalslim8936
2 жыл бұрын
Any book recommendations?
@antonvakhitov9477
5 ай бұрын
I use "Introduction to Lie Algebras and representation theory" by James Humphreys for my studies. It is a way of thinking about Lie algebras from the perspective of linear algebra and not diff geometry, but it is quite a nice book
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