Horrible sound; I downloaded it and applied VLC equializer but it doesn't really help. I wish someone would filter it with more spohisticated tools....
I have learned a great deal from this lecture. At 11:58, If I remember correctly, the Lie bracket of two Lie algebras should be " the linear combination " of elements of these sets.
If all of this seems hopelessly abstract, do not despair, he spends the entirety of the next two lectures going over everything to date, including this material, as applied to a single example. (It's still pretty abstract, though.)
Coming from Mathematics and taking a course on Lie Algebras in general, let me tell you that I watch this video at the moment, because it's not so Abstract. He even chooses a basis...
He's dropping a lot of proofs though. The characterization of Semi simple Lie algebras is enough to fill a quarter of a lecture (why does these hold). Same goes for solvability
1:14:44 Shouldn't it be that Weyl group be defined as the group generated by W, not just W? I was confused when I apply his definition to sl(2,C) treated in lecture 16, and the only way to be consistent seems to define Weyl group as the way I mentioned.
1:05:04 when he says "that means this guy is an element of H*" I had to remember that H* is linear combinations of elements in π with complex coefficients, and π is a subset of a set of eigenvalues of the Lie bracket of complex vector spaces, so the elements of π are complex. So it makes sense even though k outputs a complex number.
In 57:58, do we actually choose only the "+"s OR only the "-"s of the roots (witch are linearly independent) so they span H*? Is this the reason why the set Φ is not linearly independent?
I am unsure about 1:31:So for the s_pi_i(pi_i) = -pi_i when i=j ? or = 3pi_i? if it is = -1(if i=j), then there is no constraint that the -2K*(...)/K*(...) to be non-negative (i.e. it can be negative, which violates that epsilon summation epsilon is either +1 or -1 (either positive integers or negative integers) ? May someone help to clarify this , thank you.
let me just clarify that for you, The condition seems to be that when "i" is equal to "j," the term "s_pi_i(pi_i)" should be equal to -"pi_i." Otherwise, when "i" is not equal to "j," it should be equal to "3pi_i." There's a mention of "-1(if i=j)," which may be specifying the condition when "i" is equal to "j." The statement mentions the constraint that "-2K*(...)/K*(...)" should not necessarily be non-negative. This constraint seems to contradict the requirement that "epsilon summation epsilon is either +1 or -1," which suggests that some elements in the sum should be positive and some negative. Hope that helps!
Because C is algebraically closed. That is every polynomial can be written as the product of monomials. That is necessary to get the splitting into a solvable and a semi simple part. This is connected to Lies Theorem and Jordans decomposition of Endomorphisms
It'll make more intuitive sense if you take a first course in Real Geometric (Clifford) Algebra. The complex structure is an abstraction hiding a real bivector (graded 0+2-vector) algebra, which is essentially the algebra for space rotations, or in Cl(3,1) the algebra for Lorentz rotations and spinors. If you always think of complex structure in such real geometric terms all mystery of why mathematicians tend to prefer complex structure will be revealed. Including the Fundamental Theorem of Algebra, which is essentially a 2D geometry theorem. All simple Lie algebras are bivector algebras.
How this definition of the Cartan subalgebra H is related to the more known definition of Cartan subalgebra: A Cartan subalgebra of a Lie algebra L is a subalgebra H, satisfying the following two conditions: (i) H is a nilpotent Lie algebra (ii) N_L(H)=H where N_L(H) is the normalized of H in L Does anyone knows?
The correct general definition is indeed what you have stated above. The definition Schuller presents is true only when L is a semisimple Lie Algebra. Here is a brief outline of the connection, without proof. Definition: An element X of a lie algebra G over a field F is called semisimple if ad(X) is diagonalisable over the algebraic closure of F. Definiton: A toral subalgebra H of a semisimple lie algebra G is a subalgebra wherein all elements of H are semisimple. It is a fact that every toral subalgebra is Abelian, and it is also a fact that ad(X) is simultaneously diagonalisable in an Abelian subalgebra (for all X in H). So if H is a toral subalgebra of G (over an Algebraically closed field, like C), ad(X) is simultaneously diagonalisable for all X in H. Then, there is a lemma, which answers the question: A subalgebra H of a semisimple lie algebra G is a Cartan subalgebra (by your definition) IFF H is a maximal toral subalgebra. You can find a proof of this in most textbooks on the subject. Really, we just need the simultaneous diagonalization for physics, so he just used this property to define CSAs, which is fine since he is looking at semisimple algebras.
I have a far greater familiarity with Ricci-Tensor calculus than I do with linear algebra so while I am understanding the formalism (albeit without practice due to the lack of problem sheets) the return to tensor notation felt extremely more comfortable to me in this specific lecture.
@@GabrieleDimaggio Nakahara just covers Lie algebras definitions from the differential geometry point of view. For contents of this video: Dynkin diagrams, Killing forms and so on... J. Humphreys Lie algebra Introduction to Lie Algebras and Representation Theory is a better option.
Is this a postgrad course? You’re introducing lots of definitions very quickly, especially at the start where you’re giving definitions within definitions.
I would imagine that it is meant for people who wish to know the story of the classifcation and the ideas involved without seeing the real hard work. He is teaching this course for a physics program after all. He leaves out all proofs but the simplest calculations (relatively of course), as well as details in the theorems themselves. I am only a Master's student but I believe I could take vigilent notes (especially due to the online nature of this lecture)and if I were to dive deeper into the mathematical details, I would have these notes and this lecture tho thank for helping me understand the big picture first.
Very well presented. But, I wonder why he does not solve some real physical problems, that would have made the lectures so very rich. He even talk about roots.
3:06 levi decomposition
18:37 adjoint
21:38 killing form
29:47 structure constants
39:52 Cartan subalgebra
48:43 roots of Lie algebra
1:12:26 weyl transformation
1:27:06 cartan matrix
1:37:02 dynkin diagrams
Is anyone else having problems with the sound?
yes, i too hear an annoying crackling sound in this particular video
Fortunately it seems to get fixed after around 1:20:00
@@SCRAAAWWW It really is fixed at around 40:00
@@Fematika Sound is fixed at 38:53
Horrible sound; I downloaded it and applied VLC equializer but it doesn't really help. I wish someone would filter it with more spohisticated tools....
It looks like a professor, it sounds like a tesla coil
I have learned a great deal from this lecture.
At 11:58, If I remember correctly, the Lie bracket of two Lie algebras should be " the linear combination " of elements of these sets.
This should be the same, by bilinearity of the bracket, right
At 10:48, there should be an equal to sign on the last line and not a subset sign for the condition of no non trivial ideals to hold
7:15, 14:40, 23:27, 29:24, 34:12, 42:08, 52:00 (why), 56:47, 1:07:39, 1:23:56, 1:30:50, 1:34:12, 1:45:15
Marvelous way of communication. How I can watch all lectures on Lie algebra??
If all of this seems hopelessly abstract, do not despair, he spends the entirety of the next two lectures going over everything to date, including this material, as applied to a single example. (It's still pretty abstract, though.)
Coming from Mathematics and taking a course on Lie Algebras in general, let me tell you that I watch this video at the moment, because it's not so Abstract. He even chooses a basis...
@@davidsuchodoll4124 I completely agree with that!
In theoretical physics, it only gets more abstract from there on though
Am I the only one watching this great lecture on Lie Algebra? I really enjoys professor Frederic P Schuller's teaching!!
No, you are not, I love his lectures too. His explanations and the whole presentation is crystal clear. Highly recommended!
Me too!
+Dau Jok Yeah, these lectures are incredible
He's dropping a lot of proofs though. The characterization of Semi simple Lie algebras is enough to fill a quarter of a lecture (why does these hold). Same goes for solvability
1:14:44 Shouldn't it be that Weyl group be defined as the group generated by W, not just W? I was confused when I apply his definition to sl(2,C) treated in lecture 16, and the only way to be consistent seems to define Weyl group as the way I mentioned.
1:05:04 when he says "that means this guy is an element of H*" I had to remember that H* is linear combinations of elements in π with complex coefficients, and π is a subset of a set of eigenvalues of the Lie bracket of complex vector spaces, so the elements of π are complex. So it makes sense even though k outputs a complex number.
In 57:58, do we actually choose only the "+"s OR only the "-"s of the roots (witch are linearly independent) so they span H*? Is this the reason why the set Φ is not linearly independent?
If lambda is a root, then minus lambda is also a root. That already shows the set is not linearly independent.
At 36:15, K maps to C, not to L.
Yes, he correct it @ 1:03:10
Sound quality is terrible
How Levi decomposition is helpful in classification ?
You reduced the whole problem to classifying solvable Lie Algebras. And there are some nice theorems about solvability and Nilpotency.
I am unsure about 1:31:So for the s_pi_i(pi_i) = -pi_i when i=j ? or = 3pi_i? if it is = -1(if i=j), then there is no constraint that the -2K*(...)/K*(...) to be non-negative (i.e. it can be negative, which violates that epsilon summation epsilon is either +1 or -1 (either positive integers or negative integers) ? May someone help to clarify this , thank you.
let me just clarify that for you, The condition seems to be that when "i" is equal to "j," the term "s_pi_i(pi_i)" should be equal to -"pi_i." Otherwise, when "i" is not equal to "j," it should be equal to "3pi_i."
There's a mention of "-1(if i=j)," which may be specifying the condition when "i" is equal to "j."
The statement mentions the constraint that "-2K*(...)/K*(...)" should not necessarily be non-negative. This constraint seems to contradict the requirement that "epsilon summation epsilon is either +1 or -1," which suggests that some elements in the sum should be positive and some negative.
Hope that helps!
Can someone recommend me a book where I can complement this lecture?
Modern Differential Geometry for Physicists by Chris J Isham (I'm guessing it's literally the book used for this class)
introduction to lie algebras erdmann
Any Graduate Texts in mathematics on representation theory
thank you for this video
I once had one of those thinking about thinking moments.
Why is it simpler to use Complex Lie Algebra to study its classification?
Because C is algebraically closed. That is every polynomial can be written as the product of monomials. That is necessary to get the splitting into a solvable and a semi simple part. This is connected to Lies Theorem and Jordans decomposition of Endomorphisms
It'll make more intuitive sense if you take a first course in Real Geometric (Clifford) Algebra. The complex structure is an abstraction hiding a real bivector (graded 0+2-vector) algebra, which is essentially the algebra for space rotations, or in Cl(3,1) the algebra for Lorentz rotations and spinors. If you always think of complex structure in such real geometric terms all mystery of why mathematicians tend to prefer complex structure will be revealed. Including the Fundamental Theorem of Algebra, which is essentially a 2D geometry theorem. All simple Lie algebras are bivector algebras.
@@Achrononmaster Do you recommend a specific book/set of notes on real geometric Clifford algebra? Thanks in advance!
How this definition of the Cartan subalgebra H is related to the more known definition of Cartan subalgebra:
A Cartan subalgebra of a Lie algebra L is a subalgebra H, satisfying the following two conditions:
(i) H is a nilpotent Lie algebra
(ii) N_L(H)=H where N_L(H) is the normalized of H in L
Does anyone knows?
The correct general definition is indeed what you have stated above. The definition Schuller presents is true only when L is a semisimple Lie Algebra. Here is a brief outline of the connection, without proof.
Definition: An element X of a lie algebra G over a field F is called semisimple if ad(X) is diagonalisable over the algebraic closure of F.
Definiton: A toral subalgebra H of a semisimple lie algebra G is a subalgebra wherein all elements of H are semisimple.
It is a fact that every toral subalgebra is Abelian, and it is also a fact that ad(X) is simultaneously diagonalisable in an Abelian subalgebra (for all X in H). So if H is a toral subalgebra of G (over an Algebraically closed field, like C), ad(X) is simultaneously diagonalisable for all X in H.
Then, there is a lemma, which answers the question: A subalgebra H of a semisimple lie algebra G is a Cartan subalgebra (by your definition) IFF H is a maximal toral subalgebra.
You can find a proof of this in most textbooks on the subject. Really, we just need the simultaneous diagonalization for physics, so he just used this property to define CSAs, which is fine since he is looking at semisimple algebras.
I have a far greater familiarity with Ricci-Tensor calculus than I do with linear algebra so while I am understanding the formalism (albeit without practice due to the lack of problem sheets) the return to tensor notation felt extremely more comfortable to me in this specific lecture.
How is Ricci Tensor calculus different from tensor calculus or linear algebra. That's litterly the same.
@@davidsuchodoll4124 To a mathematician, yes.
@ 1:02:40 Killing Form maps to C not to L
+Kisfox he corrected it
Sir, can you tell me which book you are following?
@@GabrieleDimaggio Nakahara just covers Lie algebras definitions from the differential geometry point of view. For contents of this video: Dynkin diagrams, Killing forms and so on... J. Humphreys Lie algebra Introduction to Lie Algebras and Representation Theory is a better option.
Sounds like all the quality was put into the lecture and none in the audio lmao. Still a great lecture though
Insanely beautiful!
I wonder I can understand what he say ...
Is this a postgrad course? You’re introducing lots of definitions very quickly, especially at the start where you’re giving definitions within definitions.
I would imagine that it is meant for people who wish to know the story of the classifcation and the ideas involved without seeing the real hard work. He is teaching this course for a physics program after all. He leaves out all proofs but the simplest calculations (relatively of course), as well as details in the theorems themselves. I am only a Master's student but I believe I could take vigilent notes (especially due to the online nature of this lecture)and if I were to dive deeper into the mathematical details, I would have these notes and this lecture tho thank for helping me understand the big picture first.
not only you,,me too
Very well presented. But, I wonder why he does not solve some real physical problems, that would have made the lectures so very rich. He even talk about roots.
Because the goal has nothing to do with physics.
1:43:55 Garrett Lisi 😂😂
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