I don't like clickbait, so I wasn't going to watch. I changed my mind when I saw the thumbnail. I was like "This is the smartest pun I ever seen". Sorry any mistakes, my english is a work in progress.
Haha. My physics professor once hosted a seminar called "Can a physicist count on CUDA?", which in Polish can be also understood as "Can a physicist count on MIRACLES?" and he was much amused when a group of theologians fell for his prank and was shocked to find themselves listen for some reason to very cryptic-sounding general-purpose computing on graphics processing units in molecular dynamics instead. 😂
This was a great light introduction to Lie Groups and Lie Algebras. It was clear, and simple. Really enjoyed it. Hope more videos on this subject are made.
Could not have described my feelings about the video better. An amazing introduction to the topic, showing the basic concepts without any complex prerequisites
Thanks for the glimpse Michael! As an undergrad standing outside the gates of some beautiful but forbidding topics, these kinds of videos are super helpful and exciting!
@5:13 "Determinant is zero" but ad-bc=1 on the board, should in fact =1 otherwise we can't get the identity matrix! And yes it really is a 3D surface embedded in 4D because of this extra constraint (can think of one of the 4 numbers as being fixed)
This is so brilliant! Lie algebras is one of these topics that felt a little bit overwhelming when I was a young physics student over a decade ago. Now, I work as a software engineer, and while the job pays the bills, I often have the feeling that it is somewhat lacking in the sense that it does not offer any real intellectual challenges. Your videos, on the other hand, bring me a sense of enjoyment of pure learning and understanding that I miss quite a lot these days!
As I was listening to this while doing something else, I had to rewind the video to look at the board and make sure that the matrices he was describing are non-singular.
Great video. You had me fooled with the title - I got a different interesting video to the one I was expecting. :) Small point. At 5:09 you claim that SL_2 is the set of matrices with determinant 0 rather than 1 (as you correctly have written down). One of those slips of the tongue which are so easy to make. It might be nice to mention that in the preface or whatever you call it, the writting above describing the video.
HOMEWORK : Let L be the real vector space R³. Define [x, y] = x × y (cross product of vectors) for x, y ∈ L, and verify that L is a Lie algebra. Write down the structure constants relative to the usual basis of R³.
SOLUTION Clearly, [,] is bilinear and anticommutative, it need only to check the Jacobi Identity: [[a,b],c] = (a × b) × c = (a.c)b − (b.c)a (Lagrange's Formula) = (c.a)b − (b.a)c + (a.b)c −(c.b)a (Inner product is commutative) = (c × b) × a + (a × c) × b (Lagrange's Formula) = [[c,b],a] + [[a,c],b] where (.) is the inner product of R³. Using the anticommutativity, 0 = [[c,b],a] + [[a,c],b] - [[a,b],c] = [-[b,c],a] + [-[c,a],b] - (-[c,[a,b]]) = [a,[b,c]] - [b,-[c,a]] + [c,[a,b]] = [a,[b,c]] + [b,[c,a]] + [c,[a,b]]. Take the standard basis of R³: d = (1,0,0), e = (0,1,0), f = (0,0,1). We can write down the structure equations of L: [d,e] = f, [e,f] = d, [f,d] = e.
I think this is my favorite video on this channel just because I’ve already been interested in manifolds for a while and learning a bit about them and this is a perfect example of how tangent spaces and Lie algebras go together so nicely.
I would love so many more videos like this on Lie algebra! I just have one question at 8:32: where does the assumption that the derivative of the path at a point must equal an element of the Lie algebra sl(2)? Who says that such derivative is guaranteed to always be embedded in the tangent space?
A very simple, accessible introduction to Lie algebras. Definitely the best I’ve seen. Starting with the visual of the group and tangent space and using the properties to determine the basis is, I think, a far clearer explanation than the standard “here’s a basis and look what we can do with it” approach. Great video 👍🏻
Very good lecture on Lie groups and Lie algebras! I've studied these for years, and this is the most comprehensible lecture I've heard on the subject. Keep it up!
I treasure vids like these that can give an intuitive intro to higher level stuff to people like me. It's been 15 years since I took a math class (calc 3) and recently I've grown a massive interest in mathematics, currently learning linear algebra on my own, and I've done lots of peeking around at basics of other higher math picking up whatever little I can bit by bit until I'm at the point where I can delve into it. Lie Algebras are one of those things that intrigued me as well, so this is fantastic.
My favorite example of a Lie group is the torus T^2 since everyone have a good picture of it as a manifold, and the group structure is really amuzing and simple
When I was in grad school, my friend and I concocted "Top 10 Lies that Math Teachers Tell Their Students". I don't remember them all, but it started with, "You can't subtract a larger number from a smaller one," "You can't divide a smaller number by a larger one," and "Negative numbers don't have square roots," and the #1 lie was, "You need to know this."
I enjoyed this little glimpse into the world of Lie algebras. Knowing that videos like this exist online is really encouraging me to self-study more advanced math topics!
Excellent and clear explanation! Thank you! I had never known what "at the identity" meant until now. There are not many Lie Algebra videos on youtube, and I don't know of any introductory ones (Richard B's are for graduate math majors). So please make more. Thank you again.
Prof. Michael u are a great insipiration to me, as a younger mathematician, u cover so much and develop in so many ways the beauty and hidden connections in math, Kudos from Peru
So very good to be talked through this intellectually beautiful topic.Recent " Bending" of math to serve needs a data science cant compare .Later on in the calm of night i'm reaching for Jacobson book in the Dover edition to study further . My own poor thought was heuristically to think of the famous bracket singling out cross products from the "squares" definig the surfaces its being a way to principle vectors or co ordinates, but i'm only an amatuer in this and pleased to be able to see the conceptual basis of Lies work set out as well as take a motivating tour of its unfoldig plus "punch" at the end where previous work pays off-Thankyou!
I've seen this presentation before in school, but I'm looking forward to where you go with it next. I remember that it's a very important algebra that's been around for a long time.
Perhaps the best short intro/overview to Lie Groups Lie Algebras ever presented on youtube. Would be real nice to expand on this with additional expositions on Lie Groups.
Came here because of the E8 Quantum Gravity stuff interested me. This is what this sounded like "The three cardinal trapezoidal formations hereto made orientable in our diagram by connecting the various points HIGK, PEGQ, and LMNO, creating our geometric configurations, which have no properties, but with location (Ohh!) are equal to the described triangle CAB quintuplicated. Therefore, it is also the five triangles composing the aforementioned NIGH - each are equal to the triangle CAB in this geometric concept! ". Lol, I need to learn more.
Once my analysis TA made us prove SL2(R) is a manifold. I had literally 0 intuition for it and thought the problem was kind of nonsensical. This was super helpful-- very clear, with great motivation!
I was thinking this better not actually be about maths greatest lie and better more be about the Lie algebras because I wanted to find out what they were! So glad that you actually went there.
This video was awesome and gave an easy introductory notion of Lie algebras. What I missed a bit was a more in-depth explanation of the bracket-relation. Does that somehow naturally arise from the definitions, i.e. can there be only one relation that fulfills this definition? Why is it the commutator relation for matrix representations? You should definitely do a followup video to discuss this relation more.
The first rule for a Lie algebra is often written [x, y] = - [y, x] (for all x, y). This is equivalent to [x, x] = 0 (for all x): Given [x, y] = - [y, x], by setting y = x we get [x, x] = - [x, x], which is only valid if [x, x] = 0. Given [x, x] = 0, by setting x = s+t we get 0 = [s+t, s+t] = [s, s] + [s, t] + [t, s] + [t, t] = 0 + [s, t] + [t, s] + 0, i.e. [s, t] = - [t, s].
I noticed the thumbnail, so I understood that "Lie" is pronounced ['laɪ] instead of ['liː], and from the pronunciation, I found that "Lie" meant a math concept based on the mathematician Lie instead of it meaning a "false statement" or "say something false to avoid undesirable situations".
4:40 It's not obvious to me how the bracket operator always gives 0 in this case. You can take any two points from the vertical line through (1,0). Say we take [(1,0), (1, 1)]. Why is it zero? I'm new to Lie algebras, so I must be missing something. Thank you.
Out of curiosity, what would I get if I took the cotangent space at the identity instead of the tangent space? What if I took the tangent or cotangent bundle over the whole Lie group? Are those structures of any use in mathematics?
This is absolutely brilliant. Something I've wanted to learn about for years, and I was tricked into learning it by the clickbait title. Well done, and brilliant presentation.
Both Lie groups and Lie algebras have applications in theoretical physics, especially particle physics. Like many group theoretic applications, this is about symmetry. It has applications in quantum mechanics and special relativity, too. See en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group, for example.
I entered once an algebraic geometry class but I could not understand the purpose of it. That's why I stick with probability and differential equations.
Can you do a video showing applications of Lie groups/algebras? I know this is not usually your wheelhouse, but at this point I feel like you are the only person who will make me understand.
Don't really understood the connection between Lie algebra and Lie group. Because as the definition says Lie group is a differentiable manifold and a Lie algebra is a tangent space to an identity on that manifold. But is there a connection between operator in Lie group and corresponding bilinear operation in Lie Algebra? How do I derive this braket [*,*] from knowing how to operate on group elements and vice versa (generally)?
5:13 "Whose determinant is zero." Board says det = ad-bc = 1; I believe what you said, not what you wrote. it won't matter since you're in the tangent space and therefore taking a path derivative of a constant on the RHS = 00.
Any DIE-HARD Math historian fan would've clicked...very elegent tutorial & tribute to the great Norwegian Mathematician Marius Sophus Lie, Father of Lie Group Theory, who ranks right up there with fellow peers Peter Ludwig Slyow & the eponymous Neils Henrik Abel...one of my all-time Math heroes!!
very fascinating and beautiful, very excited to do abstract algebra and learn about all these things when the time comes (a couple years down the road)
I'm going through Lee's Smooth Manifolds and I would really love to get more examples of lie algebras! Coming from a mostly quantum physics background, these things are very interesting to me.
After watching this, I thought that this is a useless and boring number game that he showed. Now that you say it's related to quantum physics, I'm intrigued.
@@elaadt it's not even just related: every observable quantity in the universe, according to QM, is represented by an element of a Lie Algebra. Typically this Lie Algebra comes from the group of symmetries of the physical system under consideration. For example the objects that correspond to the spin of a fermion (one per component) come from the Lie Algebra of the special linear group SL(2) of complex matrices with unitary determinant, which in turn is about rotational symmetry as you may expect from the name "spin".
How did you get the expression for the bilinear operation on your Lie algebra? Did you just define one that worked? Are they unique (upto multiplication by a constant)?
I believe he was using the commutator [A,B] := AB-BA, which, for matrices is guaranteed to have the required properties. If I recall correctly (I did a single masters level course on Lie Groups 5 years ago), the commutator is regarded as the "standard" Lie-bracket.
smartest clickbait in youtube history
the toughest too :)
Yeah I was like ‘you got me 🤷’ and watched the full thing
I don't like clickbait, so I wasn't going to watch. I changed my mind when I saw the thumbnail. I was like "This is the smartest pun I ever seen".
Sorry any mistakes, my english is a work in progress.
Yessss!!
No kidding, I thought he was going to sing "No Myth."
"What we're doing here is fairly simple."
LIE
groups
Haha.
My physics professor once hosted a seminar called "Can a physicist count on CUDA?", which in Polish can be also understood as "Can a physicist count on MIRACLES?" and he was much amused when a group of theologians fell for his prank and was shocked to find themselves listen for some reason to very cryptic-sounding general-purpose computing on graphics processing units in molecular dynamics instead. 😂
Cholera jasna :3
A niech to dunder świśnie.
Niech to licho!
thats utterly hilarious
I'm sending this to my pastor because he will love it. Hilariously insidious.
I clicked specifically because I was hoping to dive into Lie groups/algebra. Not disappointed. I'd love to see more.
The lie is a Lie.
This was a great light introduction to Lie Groups and Lie Algebras. It was clear, and simple. Really enjoyed it. Hope more videos on this subject are made.
Could not have described my feelings about the video better. An amazing introduction to the topic, showing the basic concepts without any complex prerequisites
we need more stuff like this on YT. There is enough of calculus and linear algebra already.
5 minutes in and I was wondering "When is he going to talk about the big falsehood?" That's what I get for watching these first thing in the morning.
My head is still twitching when he said determinant is 0 when the chalkboard reads 1 at 5:10. So you do get quite a big falsehood there.
@@s4623 He meant to say 1.
There’s a reason why in the title says the word “Lie” is capitalised 😉
@patrickyip8554 The Lie group theory was given by the scientist Sophus Lie. That's why capitalised.
Thanks for the glimpse Michael! As an undergrad standing outside the gates of some beautiful but forbidding topics, these kinds of videos are super helpful and exciting!
@5:13 "Determinant is zero" but ad-bc=1 on the board, should in fact =1 otherwise we can't get the identity matrix! And yes it really is a 3D surface embedded in 4D because of this extra constraint (can think of one of the 4 numbers as being fixed)
This is so brilliant! Lie algebras is one of these topics that felt a little bit overwhelming when I was a young physics student over a decade ago. Now, I work as a software engineer, and while the job pays the bills, I often have the feeling that it is somewhat lacking in the sense that it does not offer any real intellectual challenges. Your videos, on the other hand, bring me a sense of enjoyment of pure learning and understanding that I miss quite a lot these days!
5:06 Don't you mean "whose determinant is 1"? (not zero?)
Of course that's what he means, that's what he wrote on the board.
When you mean to say the multiplicative identity but you say the additive identity
As I was listening to this while doing something else, I had to rewind the video to look at the board and make sure that the matrices he was describing are non-singular.
Now I'm having to go back to look at your older videos to understand stuff. Fantastic!
1) When I saw the title, I assumed this is where you were heading, and 2) Yes, please do more videos on this topic.
Looks so interesting, i would love to see more about Lie algebras and Lie groups
Great video. You had me fooled with the title - I got a different interesting video to the one I was expecting. :)
Small point. At 5:09 you claim that SL_2 is the set of matrices with determinant 0 rather than 1 (as you correctly have written down). One of those slips of the tongue which are so easy to make. It might be nice to mention that in the preface or whatever you call it, the writting above describing the video.
I came here to make the point about the determinant 🙂 Thanks
I got clickbaited, then saw the words "lie algebra", and now I'm gonna stay in your channel forever
Quantum Mechanics Professor: We can see its just the Lie group-Lie algebra correspondence.
Class: What??
HOMEWORK : Let L be the real vector space R³. Define [x, y] = x × y (cross product of vectors) for x, y ∈ L, and verify that L is a Lie algebra. Write down the structure constants relative to the usual basis of R³.
SOLUTION
Clearly, [,] is bilinear and anticommutative, it need only to check the Jacobi Identity:
[[a,b],c] = (a × b) × c
= (a.c)b − (b.c)a (Lagrange's Formula)
= (c.a)b − (b.a)c + (a.b)c −(c.b)a (Inner product is commutative)
= (c × b) × a + (a × c) × b (Lagrange's Formula)
= [[c,b],a] + [[a,c],b]
where (.) is the inner product of R³.
Using the anticommutativity, 0 = [[c,b],a] + [[a,c],b] - [[a,b],c] = [-[b,c],a] + [-[c,a],b] - (-[c,[a,b]]) = [a,[b,c]] - [b,-[c,a]] + [c,[a,b]] = [a,[b,c]] + [b,[c,a]] + [c,[a,b]].
Take the standard basis of R³: d = (1,0,0), e = (0,1,0), f = (0,0,1). We can write down the structure equations of L: [d,e] = f, [e,f] = d, [f,d] = e.
@@goodplacetostop2973 bruh why this video has less views , is I a part of time traveling now
@Lakshya Gadhwal And you would be wrong
@@goodplacetostop2973 I love you and your channel 😁
@@NoNTr1v1aL Thanks, my friend!
I think this is my favorite video on this channel just because I’ve already been interested in manifolds for a while and learning a bit about them and this is a perfect example of how tangent spaces and Lie algebras go together so nicely.
I would love so many more videos like this on Lie algebra! I just have one question at 8:32: where does the assumption that the derivative of the path at a point must equal an element of the Lie algebra sl(2)? Who says that such derivative is guaranteed to always be embedded in the tangent space?
A very simple, accessible introduction to Lie algebras. Definitely the best I’ve seen. Starting with the visual of the group and tangent space and using the properties to determine the basis is, I think, a far clearer explanation than the standard “here’s a basis and look what we can do with it” approach.
Great video 👍🏻
Who thought we’d all be clikckbaited into videos about Lie groups.
I’m not even mad.
Very good lecture on Lie groups and Lie algebras! I've studied these for years, and this is the most comprehensible lecture I've heard on the subject. Keep it up!
I treasure vids like these that can give an intuitive intro to higher level stuff to people like me. It's been 15 years since I took a math class (calc 3) and recently I've grown a massive interest in mathematics, currently learning linear algebra on my own, and I've done lots of peeking around at basics of other higher math picking up whatever little I can bit by bit until I'm at the point where I can delve into it. Lie Algebras are one of those things that intrigued me as well, so this is fantastic.
My favorite example of a Lie group is the torus T^2 since everyone have a good picture of it as a manifold, and the group structure is really amuzing and simple
Fascinating stuff, do more ! Also do more topology !
Finally! Yep, I would like to see other videos about this topic.
3:02 G2 Esports
15:02 The cake is a Lie
When I was in grad school, my friend and I concocted "Top 10 Lies that Math Teachers Tell Their Students". I don't remember them all, but it started with, "You can't subtract a larger number from a smaller one," "You can't divide a smaller number by a larger one," and "Negative numbers don't have square roots," and the #1 lie was, "You need to know this."
These sound like something a number theorist would say.
You wasted time.
I enjoyed this little glimpse into the world of Lie algebras. Knowing that videos like this exist online is really encouraging me to self-study more advanced math topics!
Excellent and clear explanation! Thank you! I had never known what "at the identity" meant until now. There are not many Lie Algebra videos on youtube, and I don't know of any introductory ones (Richard B's are for graduate math majors). So please make more. Thank you again.
I got the joke but I clicked anyway cause the clickbait was too good not to
was just about to write an angry comment regarding the clickbait ...
... then I realized the true meaning of the „Lie“ in your title. Now I love it!
Loved this video. Lie groups/algebras are one my favorite topics. Thanks for the trip down memory lane!
Yes to more Lie Algebra
Prof. Michael u are a great insipiration to me, as a younger mathematician, u cover so much and develop in so many ways the beauty and hidden connections in math, Kudos from Peru
This may be the best video that you have created - and you offer very tough competition to yourself. Thank you.
I have been wanting to learn about lie groups for a while now.
Same, and this was so simple and well explained, why couldn't this have been done in my undergrad classes?
So very good to be talked through this intellectually beautiful topic.Recent " Bending" of math to serve needs a data science cant compare .Later on in the calm of night i'm reaching for Jacobson book in the Dover edition to study further . My own poor thought was heuristically to think of the famous bracket singling out cross products from the "squares" definig the surfaces its being a way to principle vectors or co ordinates, but i'm only an amatuer in this and pleased to be able to see the conceptual basis of Lies work set out as well as take a motivating tour of its unfoldig plus "punch" at the end where previous work pays off-Thankyou!
My brother in law got his PhD on Lie Fields. I sent him a copy of this video. He's gonna love it.
I've seen this presentation before in school, but I'm looking forward to where you go with it next. I remember that it's a very important algebra that's been around for a long time.
these videos are an amazing resource, clearest explanation I've ever seen of Lie algrebra for a non-mathematician! Thanks so much!
Perhaps the best short intro/overview to Lie Groups Lie Algebras ever presented on youtube. Would be real nice to expand on this with additional expositions on Lie Groups.
Came here because of the E8 Quantum Gravity stuff interested me. This is what this sounded like "The three cardinal trapezoidal formations hereto made orientable in our diagram by connecting the various points HIGK, PEGQ, and LMNO, creating our geometric configurations, which have no properties, but with location (Ohh!) are equal to the described triangle CAB quintuplicated. Therefore, it is also the five triangles composing the aforementioned NIGH - each are equal to the triangle CAB in this geometric concept! ". Lol, I need to learn more.
Once my analysis TA made us prove SL2(R) is a manifold. I had literally 0 intuition for it and thought the problem was kind of nonsensical. This was super helpful-- very clear, with great motivation!
I was thinking this better not actually be about maths greatest lie and better more be about the Lie algebras because I wanted to find out what they were! So glad that you actually went there.
Your videos are awesome! Never stop making them! Very clear explanation of what Lie groups are.
i would like more of this type of video. this is also a good prereq for your VOA videos
I took a whole 24 lecture course on Lie algebras a few years ago. I've forgotten all of it... But I actually remember some of this stuff!
Same here.... over 20 years ago....
Can you suggest some good resources over the web on lie groups ,please.
Thanks!
Absolutely amazing video!
I really, really enjoyed this introduction to the subject and would love future videos. Thanks Michael!
This video was awesome and gave an easy introductory notion of Lie algebras. What I missed a bit was a more in-depth explanation of the bracket-relation. Does that somehow naturally arise from the definitions, i.e. can there be only one relation that fulfills this definition? Why is it the commutator relation for matrix representations? You should definitely do a followup video to discuss this relation more.
I liked it very much that you defined the path gamma and not just wrote identity = identity + epsilon H + O(epsilon^2) with epsilon
Well this was indeed nice. I'd like more of these.
The first rule for a Lie algebra is often written [x, y] = - [y, x] (for all x, y). This is equivalent to [x, x] = 0 (for all x):
Given [x, y] = - [y, x], by setting y = x we get [x, x] = - [x, x], which is only valid if [x, x] = 0.
Given [x, x] = 0, by setting x = s+t we get 0 = [s+t, s+t] = [s, s] + [s, t] + [t, s] + [t, t] = 0 + [s, t] + [t, s] + 0, i.e. [s, t] = - [t, s].
Yes, finally. More group teory, more examples. I hope some day we will look on all groups families (including sporadic groups).
I noticed the thumbnail, so I understood that "Lie" is pronounced ['laɪ] instead of ['liː],
and from the pronunciation, I found that "Lie" meant a math concept based on the mathematician Lie
instead of it meaning a "false statement" or "say something false to avoid undesirable situations".
4:40 It's not obvious to me how the bracket operator always gives 0 in this case. You can take any two points from the vertical line through (1,0). Say we take [(1,0), (1, 1)]. Why is it zero? I'm new to Lie algebras, so I must be missing something. Thank you.
Love your videos. Brilliant accent as well! Ive learned more than a few things from your obvious expertise.
Out of curiosity, what would I get if I took the cotangent space at the identity instead of the tangent space? What if I took the tangent or cotangent bundle over the whole Lie group? Are those structures of any use in mathematics?
Yes! Please more basic stuff like that! Maby some variational calculus in context of field theory - so with tensors and such?
Excellent lecture and so interesting! I hope for more on this subject! Also I'm curious how Lie algebras relate to differential forms.
This is absolutely brilliant. Something I've wanted to learn about for years, and I was tricked into learning it by the clickbait title. Well done, and brilliant presentation.
Yes, please, more of lie algebra! I find it interesting but sometimes it is hard to get into
Good video. You should make more of them.
This is a great introduction into Lie algebras. How are they used?
Both Lie groups and Lie algebras have applications in theoretical physics, especially particle physics. Like many group theoretic applications, this is about symmetry. It has applications in quantum mechanics and special relativity, too. See en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group, for example.
Thanks for this. I've always wanted to know a little about Lie groups/algebras and now I do. I'd love it if you did more.
Please do more of this! It will be useful for many physicist
8:38 Can someone help me understand why γ'(0) should be an element of G and be equal to…
[x y
z w] ?
I'm actually surprised at how much of this I was able to follow.
I knew the title was a clickbait, but even knowing that, I clicked it
Very nice video!! Wasn't expecting to finish it fully but it seems really interesting.
I entered once an algebraic geometry class but I could not understand the purpose of it. That's why I stick with probability and differential equations.
Can you do a video showing applications of Lie groups/algebras? I know this is not usually your wheelhouse, but at this point I feel like you are the only person who will make me understand.
Its mad how i can look at a subject not understand a word & then watch a yt video that makes it feel somewhat easy…
Your ability making very complex concepts slip so easily into my brain will end up blowing my mind! 😄
Thank you very much 👏👏👏👏
I really enjoyed watching this introduction. Hopefully, there will be next video on Lie groups and algebras
Dude after 1 minute of listening I already know you are a good mentor.
i would like too see more videos on this topic, it seems very interesting
I want more on Lie algebras pls
Don't really understood the connection between Lie algebra and Lie group.
Because as the definition says Lie group is a differentiable manifold and a Lie algebra is a tangent space to an identity on that manifold. But is there a connection between operator in Lie group and corresponding bilinear operation in Lie Algebra? How do I derive this braket [*,*] from knowing how to operate on group elements and vice versa (generally)?
I was expecting something from lie algebra but still clickbaited but it's fine
I mispronounced Lie for a long time before I heard it as "lee." Same with some other names. One of the dangers of studying solo.
Any mathematician's name that you've read but you've never heard? You're almost certainly saying it wrong.
You-ler
I absolutely adore the title.
I can't get enough algebra. Keep it coming.
This was such a clear introduction, well done!
great intro to the topic. might also be helpful to talk about where such math is used, and what its significance is.
5:13 "Whose determinant is zero." Board says det = ad-bc = 1; I believe what you said, not what you wrote. it won't matter since you're in the tangent space and therefore taking a path derivative of a constant on the RHS = 00.
Wow I loved the topic!! Please do more videos like this!
Would love to see more videos on lie groups and lie algebras
Very nice video
Hoping that you will continue your abstract algebra playlist and do advance abstract algebra like sylows theorem etc .
You done great work everyday. I like your videos and channel .
Micheal Penn is slowly becoming a real UA-camr
Any DIE-HARD Math historian fan would've clicked...very elegent tutorial & tribute to the great Norwegian Mathematician Marius Sophus Lie, Father of Lie Group Theory, who ranks right up there with fellow peers Peter Ludwig Slyow & the eponymous Neils Henrik Abel...one of my all-time Math heroes!!
very fascinating and beautiful, very excited to do abstract algebra and learn about all these things when the time comes (a couple years down the road)
Another really interesting video. Thanks Michael!
I'm going through Lee's Smooth Manifolds and I would really love to get more examples of lie algebras!
Coming from a mostly quantum physics background, these things are very interesting to me.
After watching this, I thought that this is a useless and boring number game that he showed.
Now that you say it's related to quantum physics, I'm intrigued.
@@elaadt it's not even just related: every observable quantity in the universe, according to QM, is represented by an element of a Lie Algebra. Typically this Lie Algebra comes from the group of symmetries of the physical system under consideration.
For example the objects that correspond to the spin of a fermion (one per component) come from the Lie Algebra of the special linear group SL(2) of complex matrices with unitary determinant, which in turn is about rotational symmetry as you may expect from the name "spin".
How did you get the expression for the bilinear operation on your Lie algebra? Did you just define one that worked? Are they unique (upto multiplication by a constant)?
I believe he was using the commutator [A,B] := AB-BA, which, for matrices is guaranteed to have the required properties.
If I recall correctly (I did a single masters level course on Lie Groups 5 years ago), the commutator is regarded as the "standard" Lie-bracket.
"I probably click-baited the title"
Mwahaha, your dz(i) mind-tricks don't work on me!
"These are not the toroids you are looking for"
“You need to go home and think about the real number system”
Definitely enjoyed! Thank you!