Lie algebras with
ฝัง
- เผยแพร่เมื่อ 9 ก.ค. 2022
- Teaching Tom Crawford a bit about my favorite subject -- Lie algebras.
Check out Part 2: • Heisenberg's Uncertain...
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Thanks for having me - I learned a lot that's for sure!
Awesome collab!
Fantastic collab
Are combs in short supply in Britain? Perhaps you and Boris could go halves in one.
most clear introduction into Lie algebra out here on youtube I think. Somehow very well connecting all those pieces from other videos. Not really basic linear algebra stuff so rather special thing. I mean from basic linear algebra do you know what is a basis of a matrix. Maybe yes but nobody tells you seperately and you end up thinking what does this mean anyway. Or how do you find those. I guess it is not really that interesting without those number like properties of matrices, meaning a product or an inverse.
Good examples :-)
Can you please show me how to visualise a lie algebra?
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!
Quid significat nomen tuum? Est-ne latina?
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!
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.
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!
Amazing! Love the visible excitement he has when presenting this topic
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.
This was fantastic, thank you
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.
This is good stuff! Please make more!
This is the first Lie Algebra that has ever made sense to me. Thanks!
Fantastic collaboration!!
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 TH-cam, 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.
What a cool concept to collaborate like this!
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.
From what I have seen the main difference stems from mathematics using the real exp function while the physicist seems to use the imaginary exponential- this because of the relationship that can be made with unitary morphisns and hermitian morphisms
While numbers start at 1 and natural start at 0 due to definition of empty set
Whole numbers blackboard bold W start at 1 natural is the cantor integers
This is why I teach myself both physics and rigorous mathematics, so that I can pick up any graduate math book and learn stuff like a math student would and learn what I need
@@johnsalkeld1088 While there is flexibility in the way Natural numbers are defined; Whole numbers start at 0. For mathematical works, the author should establish the convention (definition) they are going to use throughout their discussion. If there is a distinction made, we probably have \mathbb{W} = \mathbb{N} \cap \{ 0 \} . Cheerful Calculations.
So cool please more of that it is fantastic
So awesome! Love both your 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
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!!
This was great, would love to see it added to the collaborations playlist on the channel
Ohhh Mr Tom your lessons are really entertaining😊thanks a lot
Really enjoyed. This. As I say, would love to have looked at Lie Algebras during my degree.
Great collab. Love these channels
8 minutes in and I'm already really enjoying this.
I wish there were more lecturers like this ...
This video got me into this series.
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 !
No, stop, that's enough. Too many lies. I can't stand it! )))
LOL!
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.
Bring a whole playlist on lie algebra
I love it when you Lie to us.
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
Fantastic! My two favourite TH-cam Math channels coming together to collaborate!
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.
Hmmm, your prof didn’t make you feel pretty damn cool for essentially figuring it out on your own? Whenever my naive observations later turned out to be correct I found that exhilarating, and not a little bit reassuring. Hoping your disappointment was not induced by the way he handled it!
In Switzerland, we learn the cross product in math at around 17 years of age (whatever class that is in the US).
Numberphile appearance when?😎
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
Googling "that one hexagonal diagram in quantum physics that I can't seem to name" eventually ends up suggesting the Lie algebra of the group SO(5) from "Quantum Mechanics: Symmetries".
@@jounik THANK YOU ITS SO(3) ITS THE COLOR CHARGE OF BARYONS AND MESONS THAT FORM A HEXAGON AND A TRIANGLE
@@lexinwonderland5741 Ah. The eightfold way.
@@jounik ☸️ speaking of, if you're a topology or algebraic geometry nerd I recommend reading The Eightfold Way about the Klein Quartic manifold. Absolutely gorgeous mathematics, it has hyperbolic heptagonal symmetry plus a few other symmetry groups, fitting the title 7+1=8fold
@@lexinwonderland5741 I'm very much a topology and algebraic geometry nerd, so much so that I'm fairly certain that the underlying symmetry in the Standard Model, SU(3)_c × SU(2)_L × U(1)_Y generated by the three charges of color, weak isospin and weak hypercharge is itself emergent, not fundamental. Leptonicity is closer to the core than that.
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 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
If you have a Lie group, such as like, matrix groups, such as “the group of all invertible n by n matrices”, or “all n by n matrices with determinant equal to one”, or “all n by n rotation matrices”,
if you want to talk about like, a smooth path in that group, then the derivatives of that path will be closely related to the Lie algebra corresponding to that Lie group.
(“Lie” is pronounced “Lee”.)
So, you know how like, if you have a finite group, you can have a collection of generators for that group? Like how 1 generates the group of integers?
The Lie algebra of a Lie group can be thought of as “infinitesimal generators” of the Lie group.
I think whenever the elements of your group are a smooth manifold, and the group operation is a smooth map, and the inverse operation is also a smooth map, then I think that is what is called a Lie group (but I haven’t checked to make sure I have the definition exactly right.)
You can describe a smooth path in a Lie group by specifying which element of the Lie algebra to use at each time.
Could you post a close-up photo of your T-shirt, along with where it's from?
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).
Amazing video, do some in galois theory
om g.. great collabo
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
26:25 Don’t know if it‘s the right one but there is the grassmann identity for triple cross products
this is amazing! great video, I learned 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 $\mathbb{N}_0$)
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!
An introduction to lie algebras by Erdmann and Wildon covers some representation theory
awesome
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?
I have loved this subject since I learned from Schafer that all associative algebras are non-associative algebras! What could be cooler?
if you could upload in 4k, that would be really niece
An associative algebra is just a ring homomorphism (possibly where the domain is required to be a field, but this isn’t super-necessary)
Great job Thanks but i found that basis vector k x j should be equal to -i not i. Is that right?
Ye
not really the point of the video but yes
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.
I believe Michael has said he plans on making a series about Lie algebras
@@schweinmachtbree1013 Nice.
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.
Yep, I first met it at A level too.
Same here, though never did quaternions in school, I'm impressed!
hai bác cùng dạy học vui thật là vui. Toán học hay đó.
"The zero is a number with an information destructive projectaform sub set morphism. ..." :D
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.
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?
"Maths not math." Noted
I have to know the story behind that red spot. My guess is bleach 😛
(great video!)
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
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.
Letting 0 be a natural number is a useful convention in this setting, but if my historical perspective is correct, zero was "unnatural" for quite some time. And I won't summon the bromide, divinity gave mankind 1,2,3, ..... and all else is manmade.
I've seen the two be distinguished by writing the natural numbers with 0 just as N, and exluding 0 as N with a subscript +
To me it makes more sense for N to include 0 by default, then you can talk about a "natural number > 0" when necessary. Adding 0 when it would otherwise be excluded requires some less eloquent wordage.
And yeah, I think "whole number" is a term invented by school textbook writers, probably in an attempt to avoid confusing the kiddies. I don't think I've ever seen a real mathematician use it.
The Lagrange product is what I think y'all mean for the triple product.
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
I think it comes down to what you mean by a "vector". In the context of physics, a 3D vector isn't just something with three components, it also has to transform a certain way under a change of spatial coordinates. Without that requirement, the distinction between vectors and pseudovectors doesn't arise.
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.
"
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]
@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?
For a brief moment I thought this is a Numberphile video
12:15 k cross j is actually -i not i
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.
The world is waiting for the Magnus climbing math crossover collab.
I am too!!
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.
we had linear algebra as one of three main topics in high school math. cross- and dotproduct is something everyone had to know from 11th grade on.... for calculating normal vectors on planes, or to get the area of parallelograms etc
Tại sao: detA=det(A^c)
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!! :)
14:08: I got see vectors in high school PreCalc.
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.
I think maybe you have to add the identity to the set of a_n, and he just forgot to mention that?
yeah, I think \alpha_0 should be the identity, because it doesn't make any sense to put the zero function into your generating set
@@Czeckie Yes lol I haven't even considered that it was defined as a generator of something
các bạn có thể giải: 4^(4^x)=10. x?
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(?)
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.
Porkamadonna
35:45 Just a subtle correction. You said 0+(-2)=0 but I believe you meant 0+(-2)=-2.
why? detA=detA^c
Greek for vectors and latin for scalars? Why? :'(
I am shocked. In Spain we see the cross product in High School, at 16/17 yo
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!
The space in QM without the nucleons is U(1)xSU(2)
So, yes...
Operators in this Hilbert space.
If you replace the scalar n in the definition of his alpha_n with -i hbar, then it is exactly the position (alpha_-n) and momentum (alpha_n) operators in the position representation for multiple spatial dimensions (en.wikipedia.org/wiki/Momentum_operator)
@@armagetronfasttrack9808 hmm OK but it's not just the momentum transformation matrix that contains the stuff we use to operate, but every one of them more or less.
Example, in QM the famous annihilation and creation operators, which get an upgrade with field operators, don't have to worry about dimensions. They are like ladder f, varying the permitted values of the parameter like in momentum, we need max and min points, in quantum we have eigen stuff, so it's more useful than they seem at a glance, commuting rulz!
Tom's plurals kill me! 😂 But he's the Oxford prof, so 🤷♂️
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.
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 love the episodes where you lose me at the simplest definition already.
Next step: Michael on @Numberphile talking about Lie Algebra...
Next next step: Michael and Tony Padilla ON @Numberphile talking big numbers extracted with Lie Algebra
Next next next step: Michael with Matt Parker building a domino system to simulate a cross product on Lie Algebra
LET'S MAKE THESE HAPPEN!
Cool how that SL(2) looks a little like angular momentum and it’s addition in QM.
I'd prefer to learn Truth Algebras
Please call the Fashion Police
cần có sự giải thích dễ hiểu.
It's all a Lie!
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.
N is not positive integers. If you want positive integers use positive integers.