Lie algebras with  

Thanks for having me - I learned a lot that's for sure!

What a treat!! I'm a physicist and I love both channels, seeing you guys together on one of my favorite topic (Lie algebra) is incredible! I hope you can make more of these cross-over!

This collaboration is helpful. As a physicist, I have been exposed to Lie algebras and would like to know more. However, the more general, rigorous treatments presented by many mathematicians are almost unrecognizable to me.

This was a delightful presentation! I love everything with Lie algebras. It was great seeing a Heisenberg Lie algebra, as that is not common in introductions. I also really appreciated the multiple nods towards representation theory and the lattice of weights.

This was a fantastic pair of videos! I love how the two topics were intertwined with each other, and how we got to share in the “aha” moments with the respective “students”. I’m a fan of Socratic learning, so I particularly liked when the student got to contribute to the next step of the proof, and I’d love to see more of that happen if you do more of these videos.

Oh man, I love how your channel grew. I loved and love your videos about competition problems, however i really LOVE that you trying to introduce some higher mathematics to us. I like to think that you might be making your viewers feel that theoretical mathematics isn't that scary and complicated and I'm here for that!

12:15 k cross j is actually -i not i

Really looking forward to this. Will watch later tonight. Just watched the QM one which I enjoyed. Interesting pairing as I did the QM in my second year at Cambridge, whereas I didn’t get an opportunity to do Lie Algebras at undergrad level. I think some years it was available in Part III but not my year. Shame, as finite simple groups were a big interest of mine and Lie Algebras is very complementary to that. The QM I studied in 1985, way before the internet, so my choices of presentation were either a not great lecturer, or some dry textbooks. So envious of today’s students who can look at several different order presentations of a subject on RU-vid, such as Binney or Susskind. Ironically, I reckon I understand QM now, despite leaving academia in 1988, than I did at the time, because of this! Keep up the good work, I follow both channels closely.

This is the first Lie Algebra that has ever made sense to me. Thanks!

So awesome! Love both your channels

This was great, would love to see it added to the collaborations playlist on the channel

What a cool concept to collaborate like this!

Incredible video. You a doing a service to human kind by putting this caliber of material out there for the public. Simply brilliant. Keep it up please and I wish you get very nice things for Christmas!

Note that in geometric algebra the cross product in 3 dim space = i (ab-ba) where i = xyz a unit volume in 3 space and ab is the product of two vectors giving an area - so the cross product is actual a hidden form of commutator.

I know this is older, but I would LOVE another collab video like this! Especially if you start talking about sl3 ;) or how su2 can be made from a real and imaginary copy of basis sl2 spaces or whatever fits in that vein. can't wait for your Lie theory series on mathmajor soon!!

Amazing! Love the visible excitement he has when presenting this topic

Great collab. Love these channels

29:00 That's the little bits of a video like that which makes it so human and so lovely!! Thank you for the whole video but I just can't not point out that hilarious moment

fantastic!! I would love to see more content on Lie groups/algebras, teasing the "weights" of sl2 reminds me of that one hexagonal diagram in quantum physics that I can't seem to name

So cool please more of that it is fantastic

8 minutes in and I'm already really enjoying this.

Fantastic! My two favourite RU-vid Math channels coming together to collaborate!

I love it when you Lie to us.

I thought you will tell more about the motivation behind the discovery of Lie algebras or why they are so important :) So far looked like neat structures in themselves

It was nice to see more of your personality in between the math bits - a nice shake up from the refined, concise presentations you typically present.

Numberphile appearance when?😎

Fantastic collaboration!!

No, stop, that's enough. Too many lies. I can't stand it! )))

I just really enjoy the Peano axioms for generating the natural numbers: Simple and clean. So for me, 0 ought to be included. The strictly positive integers are then usually notated as Z^+ But ultimately, yeah, it hardly matters as long as you are clear about which you mean

Ohhh Mr Tom your lessons are really entertaining😊thanks a lot

Idk about other parts of Europe, but in Switzerland the natural numbers don't automatically include zero. If you want to refer to the natural numbers including zero, you write it as the blocky N natural number symbol with a zero in the subscript (In LaTeX you'd write that as $\mathbb{N}_0$)

I wish there were more lecturers like this ...

This is good stuff! Please make more!

This was fantastic, thank you

An associative algebra is just a ring homomorphism (possibly where the domain is required to be a field, but this isn’t super-necessary)

In Switzerland, we learn the cross product in math at around 17 years of age (whatever class that is in the US).

Why does an algebra need to be defined with a vector space? Essentially, can we say that an algebra is a vector space with an additional operation, namely multiplication? A vector space itself has only two operations: addition and scalar multiplication. By introducing multiplication, we enhance the structure to form an algebra.

Bit surprised you both feel that you don’t get introduced to dot and cross products until university/college. I had to do it as part of A Levels, and using quaternions, so we were well aware of the loss of associativity. Admittedly there was none of the formality around fields/rings and algebras, was more around having tools to solve particular problems in vector space.

Dear Michael, I really enjoyed your video, it was great. If possible, please make a video about clifford algebra and group theory. Thank you very much

Hi, I just got that we are taking about polynomial transformations and not polynomials themselves. Very nice teamwork and application of the Heisenberg's Uncertainty Principle !

Alright, but I'm enjoying your class because I'm kinda old and I already know these maths, but not much resources like this back then... Good job

26:25 Don’t know if it‘s the right one but there is the grassmann identity for triple cross products

"The zero is a number with an information destructive projectaform sub set morphism. ..." :D

Edgar F. Codd invented an algebra to do with database tables. They have inner joins and outer joins and they also use a cross product join sometimes. That is every record to record combination of the two tables.

17:05 Exercise: "=>": Let b = a. We have [a, a] = -[a, a] => 2[a, a] = 0. Assuming char(L) != 2, it follows [a, a] = 0. "

This video got me into this series.

Really enjoyed. This. As I say, would love to have looked at Lie Algebras during my degree.

Bring a whole playlist on lie algebra

Great job Thanks but i found that basis vector k x j should be equal to -i not i. Is that right?

om g.. great collabo

Could you post a close-up photo of your T-shirt, along with where it's from?

If possible, could you make more content on Lie Algebras and Lie Groups? Richard E. Borcherds and a few other people have series on Lie Algebras/Lie Groups, but they're either unfinished or they don't cover topics like how to use Lie Algebras to solve differential equations, how they apply to Group Theory, etc.

Would love more videos like this. The representation theory of lie algebra looked particularly interesting ... can anyone recommend a good book on this? Thank you!

Maybe we can make that dancing AxA vector fill in a planer disk rather than getting cancelled out. So, the cross product in three dimensions is orthogonal to the two vectors. We could consider a vector as a disk with an infinite ray on one side, and the magnitude of the vector as the radius of the disk. Then the cross product would produce another ray along the intersection of the two disks (using the right-hand rule), and the radius of the resultant disk would be ||A|| ||B||. Now when we perform AxA, that which intersects are the entire disks, and the rays themselves. So, the disk is still there with a radius of ||A||^2, with the 'new' ray, the same as A. Still one problem though -- what do we do with (-A)xA ? It seems there is some ambiguity as to which way the ray would point. I think we could replace the word 'radius' with 'area' and get a workable analysis either way. I have a feeling that there is something in Geometric Algebra that goes into that(?)

I'm assuming that with your example of the cross product as a Lie algebra we're ignoring all the fuss about vectors vs. pseudovectors and sort of sweeping them under the rug :p

In Germany we we learn about the cross product in high school. There are 2 semesters about calculus, 1 about linear algebra (where the cross product is one of the topics) and 1 about statistics.

Just noting that, in the UK, the cross product is introduced to college (Sixth Form) students at pre-U level, as part of A level 'Further Maths' (currently FP1 in the post-2017 syllabus).

if you could upload in 4k, that would be really niece

Having feet in both maths and comp sci camps, it makes most sense for 0 to be included in the natural numbers. I guess 0 can be used as a subscript on the natural numbers symbol to emphasise 0 included. My school textbooks distinguished between natural - excluding zero - and whole - including - but I’ve not seen another source since that advocates this.

14:08: I got see vectors in high school PreCalc.

Maybe a bit of a long shot to expect this to be answered by but I'll try anyhow! I work with a bunch of people (mainly physicists and graphics programmers) who have a particular take on lie algebras, which is that they the *axes* of the transformations in a "spin group". The spin group is the handedness-preserving transformations within the double cover of an orthogonal group, the totality of which is a Clifford algebra. My question is: is this a way people studying lie groups often see them? If not, are there any clear drawbacks to it? I'll give some examples. Example 1: the transformations of 2D euclidean space is Cl(2,0,1). This has reflections, rotations, translations, and glide reflections. If you look at only the handedness-preserving transformations you have only rotations and translations. If you look at the axes of those transformations, you get points in 2D space together with points at infinity/"on the horizon" (a translation is a rotation around a point at infinity). Example 2: the transformations of 3D hyperbolic space is Cl(3,1,0). This has hyperbolic reflections, some rotations, hyperbolic translations, rotoreflections, hyperbolic glide reflections, and hyperbolic screw motions. Handedness-preserving transformations: rotations, hyperbolic translations, and hyperbolic screw motions. Axes of those transformations: lines in 3D hyperbolic space. A nice thing is that the axes are always the "bivectors" of the Clifford algebra, and the handedness-preserving transformations are all in the "even subalgebra" of the Clifford algebra, eg the bivectors, quadvectors, scalars ("0-vectors"), and linear combinations thereof.

At 50:48, the commutator is constant (not zero). Does that element belong to the Heisenberg Lie algebra? If it is not, then the algebra doesn't seem to be closed?

around 8.50 ish you say something like "if you want to be super fancy pants you could take something like all linear transformations from a vector space to itself" and can we get a SUPER FANCY PANTS merch with some linear transformations on them PLEASE!! :)

Porkamadonna

Amazing video, do some in galois theory

The world is waiting for the Magnus climbing math crossover collab.

awesome

Very interesting presentation - I wondering you guys are familiar with hestenes formulation of vector algebra where the general product of two vectors is a scalar plus a bivector it is really interesting and allows for reformularias of relativity and quantum mechanics and ajusts many of the familiar rules

I remember as a student in the 2nd or 3rd semester I came across SL2 and suddenly thought - hey that looks like x-Product. I worked out all the details over night, looked in 2-3 books about linear algebra which where available at that time in order to find out if this was known and I couldn't find it. Then I went to my math professor who still is quite a luminary in Lie-Algebras (he wrote a couple of books about that subject) and I found out this was a long known connection :( . You just reminded me of that sad epsiode. But anyway I still think, that was not too bad for a physics student and it was quite fun working it out by myself.

@MichaelPennMath I apologize for asking a dummy question. The brackets [ , ]: A x A -> A, but in the Heisenberg example the bracket [a_n, a_{-n}] = n = n * 1, but, I guess, 1 is not an element of A. Is it a problem?

Hi, I am a bit confused with your last example. When you take the Lie bracket of 2 opposite elements of your basis and you get a multiple of the identity... which does not seem to be in the linear span of the \alpha_n ! Shouldn't you take \alpha_0 = Id ? This wouldn't mess the brackets with the other operators. Anyway, keep up the good work! I love this format.

"Maths not math." Noted

I have to know the story behind that red spot. My guess is bleach 😛 (great video!)

also, how does the Heisenberg algebra shown relate to the matrix group/algebra associated with the 3D nilmanifold? group matrix [1,a,b;0,1,c;0,0,1]

The Lagrange product is what I think y'all mean for the triple product.

I am shocked. In Spain we see the cross product in High School, at 16/17 yo

For a brief moment I thought this is a Numberphile video

I love the episodes where you lose me at the simplest definition already.

this is amazing! great video, I learned a lot.

Cross products of vectors are in a lot of high school precalculus textbooks though I feel they are not often actually taught in the classroom, since as you say they’re not terribly important and linear algebra is not strongly valued in our typical programs where calculus is viewed as king and statistics closely behind. Like some of the finer points of the complex plane, polar functions, and series, there are a lot of topics which are nice to cover but could wait until they are ‘really needed’ in Calc 2 and beyond, since students that get that far can probably pick up the topic quickly (or so the educators in charge rationalize). The recently published AP Precalculus program (which will launch nationally next year for the May 2024 exams) has a broad scope but I didn’t see dot products or cross products on there either.

N is not positive integers. If you want positive integers use positive integers.

35:45 Just a subtle correction. You said 0+(-2)=0 but I believe you meant 0+(-2)=-2.

Do all math research guys have such well trained bodies? The wizard looking dude on youtube who makes short videos as well. (Forgot his name)

I can see hints of connections with QM, but nothing that fits exactly. a(-n) and a(n) look very similar to position and momentum operators. Interpreted that way, as expected they all commute with each other for different particles, but the commutator for the position and momentum of the same particle has a factor of n that doesn't make sense (and is also missing a factor of i). That factor of n suggests a connection with creation and annihilation operators, but if there is one, it's not as simple as identifying a(n) as a creation operator and a(-n) as an anihilation operator or vice versa. It would be interesting to hear an explanation of how exactly this all relates to QM from someone who knows what they're talking about!

I'd prefer to learn Truth Algebras

What does a generic element of C[x_0, x_1, x_2,… ] actually look like? Are they all still polynomials with finite terms? But each of the terms is a monomial over a finite subset of the variables?

But what’s defined at the end is not even an algebra. This can be fixed by taking \alpha_0 = c id, with some nonzero c. Still the result is a Heisenberg algebra only in one dimension. The definitions of \alpha’s should also be fixed. Say by dropping the factor n.

36:04 The argument of combining e with itself would lead to a weight of 4, which does not exist in that string structure, does not look convincing to me. Analogously, one could say that combining h with itself would lead to a weight of 0, which indeed exists in that string structure.

Tom's plurals kill me! 😂 But he's the Oxford prof, so 🤷‍♂️

I didn't get how sl2 and the cross product are just a "change of basis away" from each other, as with sl2 we have showed that [h,e]=2e but in terms of vectors in C^3, no vector perpendicular to itself. Meaning there are no u,v in C^3 s.t. u X v = av for some number a. What am I missing? Hopefully it is not too stupid of me haha

Greek for vectors and latin for scalars? Why? :'(

Please call the Fashion Police

Tại sao: detA=det(A^c)

It's all a Lie!

c_1(a*b) = (c_1 a)*b = a*(c_1 b) is called homogeneity.

Im annoyed that i was recommended this because ive been asking chat gpt about lie algebra, but havent otherwise indicated my interest in it.

I remember when I learned calculus and linear algebra, and felt so proud because I naivily thought that I've learned something people research about, but then I found out that it was considered "baby maths" as all mathematicians should be able to do it in their sleep. Recently I've been studying some Lie algebra, so it was cool to find out through this video that I've come to a point where the stuff I study isn't longer trivially known by all mathematicians.

Cool how that SL(2) looks a little like angular momentum and it’s addition in QM.

Physics's girl's brother?

cần có sự giải thích dễ hiểu.