Grad student here. I know a lot about associative algebras and Lie algebras, but I'd never actually heard of a Poisson algebra before. It was nice to learn a little something new today!
@tomkerruish2982
4 жыл бұрын
It shows up in classical mechanics in the Hamiltonian picture. It corresponds (in the limit hbar goes to 0) to the quantum commutator of two operators divided by i hbar.
@kejineri3022
4 жыл бұрын
It also comes up in integrable systems
@davidz5525
4 жыл бұрын
Yooooo. First year undergrad here. I’m surprised how much I can actually follow, and I definitely hope you would continue the series!
@roboto12345
4 жыл бұрын
School student here and I just read about assoviative algebras, and this is mindblowing for me. However I want to be a mathematician and I enjoy mindblowing things, so thanks
@adilattar1492
4 жыл бұрын
You were supposed to tell us "why you would want this thing to exist in the first place" at the end but you didn't end up doing it lmao. You can't leave us hanging like that. Also, I really enjoyed this, please continue putting more of these up
@thanderhop1489
4 жыл бұрын
The joke is there was no reason to want it to exist...hmmm...
@someguy3335
4 жыл бұрын
Clifehangers in math videos... wtf
@sirlight-ljij
4 жыл бұрын
He kind of did, the reason why it exists, the motivation to define a special structure is because it generalises several other algebraic stuctures
@someguy3335
4 жыл бұрын
But there are usually different ways to generalize a given concept. Why is this the chosen one?
@sirlight-ljij
4 жыл бұрын
@@someguy3335 It does not mean vertex algebras is the way to generalise. The motivation is usually a way to reach a tough concept from some well-established concepts; a location in the world of mathematics if you will.
@wesleysuen4140
4 жыл бұрын
8:42 jacobi identity on the board: the [x,z] should be [z,x]
@jkid1134
4 жыл бұрын
THANK YOU! I was proving the identities for the example problem and was stumped!
@axemenace6637
4 жыл бұрын
Thank you ... This mistake cost me many hours :(
@stephendavis4239
3 жыл бұрын
I just noticed the faux pas myself: [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0
@Kapomafioso
3 жыл бұрын
Spot on. The thing about Jacobi identity is that you just shift the elements x,y,z cyclically, so z comes to the front, x, y and then y comes to the front, z, x: [x, [y,z]] + cyc. = 0. Same thing holds for a cross product of 3 vectors in 3D, so it's not that surprising.
@littlenarwhal3914
3 жыл бұрын
thank god for this comment ive been staring at my sheet for ages
@ianmathwiz7
4 жыл бұрын
I would love to see more of this series.
@eschudy
4 жыл бұрын
awesome. keep them coming, love to see this turn into a series!
@jcreedcmu
4 жыл бұрын
Really enjoyed this and would love to see more! Especially more detail as to what "almost" being an assoc/commut algebra and Lie algebra actually entails & how the intuition you seem to be alluding to of creating particles in a Fock space relates to actual annihilation/creation operators.
@charlottedarroch
4 жыл бұрын
I remember my class on Lie algebras in university, but I appreciate it a lot more now knowing it sits in a more general framework of algebras over fields in this way. I had also never heard of a Poisson algebra, but it looks very neat. I'd especially like to hear more about the quotient algebras of the sum of tensor powers by a subalgebra. And I'd love it if you made more videos on vertex algebras!
@lenoel7666
4 жыл бұрын
The second best Cliffhanger in the history of maths. Right after Fermat.
@twistedsector
4 жыл бұрын
Awesome, I've been interested in this for a long time (mostly because of its applications to string theory and monstrous moonshine)
@johnstroughair2816
4 жыл бұрын
Fascinating lecture, I’m really looking forward to the next one.
@uvmkr
4 жыл бұрын
Please continue this series. Really liked it...👍
@robertschlesinger1342
2 жыл бұрын
Excellent video. Very interesting, informative and worthwhile video. Can hardly wait to finish this sequence of videos.
@xcheese1
4 жыл бұрын
It is SO WEIRD seeing all of this out of it's physics context, like that feeling you get as a kid seeing your teacher at the grocery store.
@ThaSingularity
4 жыл бұрын
YEESSSSSS. I really hope you continue this series
@droro8197
2 жыл бұрын
Michael, you are are awesome! I also have a MSc in mathematics and never heard about vertex algebra. I like this format of mini-course about vertex algebras - unlike many math channels choose a niche topic and explain it in a 10-20 minutes video. this is the way.
@graviton9993
4 жыл бұрын
A great introduction to the topic! I’m looking forward to the continuation of this series!
@ussenterncc1701e
4 жыл бұрын
This is wonderful, I hope you keep going with this. I'm a physics grad student, the introduction I got to these algebras was application specific; and they were not developed, I was simply told the properties of them. It is really helpful to see the motivation behind these algebras; I'd much rather understand how and why something works than memorize its workings (that's why I went into physics, lol).
@vikaskalsariya9425
4 жыл бұрын
I fricking love this! Just watched the 2nd part and rewatching again.
@paulkohl9267
4 жыл бұрын
Awesome, thanks for the gentle introduction to a topic that I have not been able to get a toe-hold onto before. This is an informative presentation that is at just the right level for someone with an intermediate knowledge of Abstract Algebra. A book on non-commutative geometry I own will hopefully make more sense now. Vertex Algebra's remind me of QFT and Fock Space.
@TheLethalDomain
2 жыл бұрын
Vertex algebras are the fundamental algebras of QFT. This is not coincidental. Associative algebras are the fundamental algebras of quantum theory and yield the canonical commutation relation. Poisson algebras yield Poisson brackets and yields the fundamental algebras of classical mechanics. Finally, Poisson Vertex algebras are the fundamental structure of Classical field theory.
@jkid1134
4 жыл бұрын
Phenomenal video, accessible and engaging.
@jeremyredd4232
4 жыл бұрын
I'm loving this so far! I hope you continue this lecture series! The applications to physics are readily apparent!
@wilderuhl3450
2 жыл бұрын
I needed this video tonight. Thank you.
@miruten4628
4 жыл бұрын
This is very interesting, please continue the series :)
@simondavenport2966
4 жыл бұрын
Really loving your videos Michael! It's awesome to see so many topics in mathematics made so accessible.This one is particuarly interesting to me as vertex algebras are used a whole lot in my field of physics, namely conformal field theory and its applications to the theory of the fractional quantum Hall effect. For a long time I've wanted to get a better handle on the underlying mathematics. You definitely have my vote for more videos on vertex algebras!
@dianeweiss4562
3 жыл бұрын
As a person who passed her physics PhD qualifying exam, but who got brushed by when I asked a few questions, I’d like your video on Rings and Ideals. This topic was not covered when I subsequently took Linear Algebra with Python, or when I tutored Linear Algebra. I used Bras and Kets in physics class, but not as operators in any space. Most of my research somehow used a Fourier Transform space for Tomographic Imaging, but I lacked mathematicians who could give me theory as to why my imposition of a +/- operator on the complex matrix resulted in clear images.
@biggbarbarian224
4 жыл бұрын
Great video! Please tell us more about this topic.
@karlwaugh30
4 жыл бұрын
That was really fascinating (as someone who has taken pure maths upto PhD) but I feel some motivating examples (& actually addressing why we would want these things) might help as an intro before all the definitions. Definitely interested in more.
@robshaw2639
4 жыл бұрын
Please continue this series!
@gianfranco2101
3 жыл бұрын
Really very interesting. I can't wait for the next one!!
@mdmishfaqahmed5523
4 жыл бұрын
nice presentation. very easy to follow.
@get2113
4 жыл бұрын
Very clear exposition. The Y got left out, but from the comments it appears these things are useful for quantum mechanics.
@nice3294
2 жыл бұрын
This looks like a very interesting topic that you're covering
@manmanbearbearpig1
4 жыл бұрын
Awesome video! Thanks for making this, can't wait to see more
@xXTobi1
4 жыл бұрын
This was really interesting, would definitely be interested in more on this topic!
@aa-lr1jk
4 жыл бұрын
Great, i was expecting this video a while since your Q.A. video. I did'n expected this topic was a somewhat main subject in research when i googled it, since i did'n even heard this before, but its really interesting to know it since i was studying a while ago tensor algebra.
@thephysicistcuber175
4 жыл бұрын
Ok, this is epic.
@lavneetjanagal
4 жыл бұрын
Very Nice video. Thank you.
@hxc7273
4 жыл бұрын
That was super interesting. Can't wait for more.
@hugolabella6417
4 жыл бұрын
Now I don't know where to stop. No place I can think of is good enough.
@jkid1134
4 жыл бұрын
Oh my goodness, the ending is hilarious! Also please do continue this :)
@declanarmstrong101
4 жыл бұрын
It's interesting to see this as the basis for so many quantum relationships. It'd be great to see what the relationship is between these generalised vertex spaces and something like a Fock space (I get the impression this is where it would be heading, but maybe that's not correct).
@ImPresSiveXD
4 жыл бұрын
Great video! I would like to know more about it but i need more background info on Lie algebra.
@willnewman9783
4 жыл бұрын
I really enjoyed the video. In another comment, you said these are related to jet bundles in algebraic geometry. It would be cool if you did an explanation of that
@goodplacetostop2973
4 жыл бұрын
So I guess good places to stop commute so they don’t belong to vertex algebra? 🤷
@35571113
3 жыл бұрын
I was looking for your comment.
@archismanrudra9336
2 жыл бұрын
Thanks! the notation in the calculation of [f, e tensor e] is a bit confusing, because you are writing this as e^2. There is a temptation to calculation the square using the form of the matrix (using the associative algebra structure) to get the zero matrix, but that is clearly not what you want there. But nit picks aside, great video, thank you
@garko123456
Жыл бұрын
there is an error in Jacobi identity of Lie algebra 6:39: second term needs to be [y, [z,x]] instead of [y,[x,z]] @Michael Penn
@twistedsector
4 жыл бұрын
could you do an in-depth video on the tensor algebra, and its quotients by ideals (UEA, exterior, Spin)?
@bonsairobo
Жыл бұрын
Typo (chalko?) in the Jacobi Identity: should be [y, [z, x]] instead of [y, [x, z]]. I spotted it because all three products are just a cyclic permutation of x, y, z.
@daveaasen
4 жыл бұрын
YES!
@omfgmx
4 жыл бұрын
Well, this is the most definetely one of the harder olympiad questions on that channel.
@MTd2
4 жыл бұрын
FIRST COMMENT, LIKE BEFORE WATCHING VIDEO! I've been your subscriber since you had less than 1k subs.
@aa-lr1jk
4 жыл бұрын
Not only you.
@chiefanand9917
4 жыл бұрын
its really cool but need more elaboration on vertex algebra especially on u_-1 and further (if possible) anyway love the video thanks
@NoYouLube
4 жыл бұрын
Two comments/questions: 1) The definition of the Lie algebra didn’t include the asymmetry [x,y] = - [y,x], but this was used later in the video. Possibly it can proved from the definition? 2) I’m not sure I understand point 3) of the definition of vertex algebras. Is a limit z -> w missing?
@hydraslair4723
4 жыл бұрын
Thanks Michael for picking up this series! Loving the general structures presented so far. But what do you mean by "nicely gluing together the associative product and the Lie bracket"? Is there a natural way to see that they blend together only if the Leibnitz rule is satisfied?
@mannyc6649
4 жыл бұрын
I don't get the locality axiom. If I apply [Y(u,z),Y(v,w)] to some vector c in V I get a formal power series in V[[z,w]]. If it is nonzero, how can that power series vanish when multiplied by (z-w)^N? Are we taking z->w? Edit: nevermind, I got it. If someone was also puzzled reply here.
@hgnb1001
4 жыл бұрын
That's a very nice cliffhanger to stop.
@sinecurve9999
4 жыл бұрын
This idea of a vertex algebra seems related to ladder operators in quantum mechanics. Creation and annihilation operators, anticommutation relations, vacuum state, etc. I've heard there's a lot of interest in Lie Algebras in mathematical physics, how about vertex products?
@joeybeauvais-feisthauer3137
4 жыл бұрын
Look up conformal field theory
@venkateshgs436
2 жыл бұрын
Prof. Penn : at 7:05 in the video, the second term of RHS of the Jacobi identity should be [y,[z,x]] ... !
@avi123
4 жыл бұрын
Thanks for introducing me to this interesting algebraic structure, I had some trouble understanding it, I on Wikipedia that it can be considered one operation V×V->V((z)) instead of infinitly many operations, can you expand on this please?
@ZeroXbot
4 жыл бұрын
V((z)) is a ring of formal Laurent series, that is infinite sum of terms a_n z^n for integer n. Note that this is exactly what the result of Y(u, z)v is and that it takes an element of VxV with z set as "constant" so it's the operation Wikipedia is describing. Now, as that series contains, as coefficients, all multiplications at once you can define u *n v by coefficient of z^(-n-1) in series you obtained by computing Y(u, z)v. So in fact you define one very big operation that encodes all those single multiplications. Hope that helps.
@urosgrandovec3409
Жыл бұрын
Based on your physical associations, this kind of reminds me of Fock space in quantum mechanics of multiple particles. If yes, could you elaborate please?
@nathanisbored
4 жыл бұрын
I think I would be interested in more but keeping it slow is a must for me. This video was already very abstract and hard for me to follow, probably just because I’ve been exposed to almost nothing discussed in this video before
@Solaris1Q84
4 жыл бұрын
Is the tensor algebra considered here even a Lie algebra? What about unique definition of terms like [ab,cd] taking bracket antisymmetry into account? So, is the considered tensor algebra a valid example of a Poisson algebra?
@SlidellRobotics
3 жыл бұрын
24:01-24:21: The logic for this implication totally escapes me.
@marcusrosales3344
2 жыл бұрын
For the tensor algebra, where would [h,e] go? It's quadratic but it is prportional to f.
@haypennyful
2 жыл бұрын
How would you go about demonstrating some of the "close" to comm/assoc of the *{-1}, for example. From properties 1 and 2 I get that it acts the right way with the vacuum element (u.1 = 1.u = u), but expanding out 3 I can't seem to prove that it is associative or commutative (or at least, not using any approach which does not imply that all other products are also assoc/comm which is obviously going down a bad road). Similarly, for *_0 I get that [1,u] = [u,1] = 0 as desired, but can't get antisym/Jacobi. Be grateful for any tips!
@RuleAndLine
4 жыл бұрын
That's a great hoodie. Can I get one?
@MrRyanroberson1
4 жыл бұрын
is it possible for there to exist some algebra which is commutative but not associative? (xy)z = (yx)z but not = y(xz) nor = x(yz)
@quantumchill5237
4 жыл бұрын
Certainly. Consider the three-dimensional algebra over a field (say, the real numbers) spanned by the set {e_1,e_2,e_3} where: (e_i)^2=0, e_1*e_2=e_2*e_1=e_3, e_2*e_3=e_3*e_2=e_1, and e_3*e_1=e_1*e_3=-e_2. It's clearly commutative, but note for instance that e_1*(e_1*e_2)=-e_2 but (e_1*e_1)*e_2=0. The algebra thus fails to be alternative as well. However every commutative algebra is flexible, which is fairly straightforward to show.
@heliceaberlin
4 жыл бұрын
20% mindblown, 80% cliffhung
@ANAND02120
3 жыл бұрын
So given the definition of vertex algebra is this a proposition following from the definition that the 0 product is close to lie algebra and -1 product is close to asso/comm alg? Is this following from axoim 3 ?
@natepolidoro4565
4 жыл бұрын
idk wtf this is but i like
@guillermodiaz2773
4 жыл бұрын
a man of culture
@bonsairobo
Жыл бұрын
The definition of VOA comes out of left field. What in the world. I think I need another video to get me there.
@bobernhardsson5345
Жыл бұрын
At 8.50 the middle term of the Jacobi identity is wrong...
@half_pixel
4 жыл бұрын
Why do we define the -1st product to be the "nice" one? Couldn't we use n instead of -n-1 in the definition of Y in order to have the 0th product be the "nice" one? That seems like it'd be a more natural definition.
@rubendepreter7402
4 жыл бұрын
The last rule (with T and partial derivative) wouldn’t be “Nice” if we used n instead of -n-1 so i think that’s the reason.
@YASHTOTLA
4 жыл бұрын
This was not very hard to follow. But I did not understand what the terms meant. Like what is a base? Different types of Algebra??? ): Can you do a different video explaining what do all the different terms mean and how do we get there? Sorry I am not well equipped to really understand what's happening here.
@TG-ge1np
3 жыл бұрын
Give us moooore! Pretty please :D
@axemenace6637
4 жыл бұрын
Is it true that in a Lie Algebra, [x,0]=0 for all x in the algebra? If not, where is the error in my proof? By Jacobi, [x,[y,x]]+[y,[x,x]]+[x,[x,y]]=0 for all x,y in G (by using z=x in the original identity). Then, by linearity, [x,[y,x]+[x,y]]+[y,0]=0. But using antisymmetry, [x,0]+[y,0]=[x+y,0]=0. We can simply let y=0 to find that [x,0]=0.
@dmitriliu5777
4 жыл бұрын
I had a quick question about the direction in which you want to take your channel: are the "course videos" (like Real Analysis for example) meant to be a listing of useful, relevant topics, or do you intend on having viewers watch them in a particular order? I was thinking of going through some real analysis notes on my own time and wanted to use the videos when I get stuck, because while your videos are amazing, I find I digest things better if I read them. If they are meant to be in a particular order, it may be useful to number them accordingly - thanks!
@MichaelPennMath
4 жыл бұрын
They are in a loose order. I use them to "flip" the classes I teach. They are not quite thorough enough to serve as the only material for a course. I fill-in details during our class meetings and leave other topics out of videos altogether, leaving them as "discovery assignments" in class.
@dmitriliu5777
4 жыл бұрын
Ah, thanks for getting back to me! I'll keep that in mind as I take the absolute plunge.
@Walczyk
4 жыл бұрын
wouldn't e (tensor product) e be a 4 dimensional matrix? so to add f to it, do we do f (tensor product) with identity? i never get this part right. How do we find these products between u and v? This is so formal o_O
@avz1865
4 жыл бұрын
Haha, I was caught off guard when it turned into quantum mechanics halfway through.
@malawigw
4 жыл бұрын
You mean the commutator?
@stevenismart
2 жыл бұрын
I want that jacket but I would prefer if the logo was slightly smaller on the back
@Czeckie
4 жыл бұрын
awesome, sad it ended in the best part!
@tomkerruish2982
4 жыл бұрын
I'm interested. This is terra incognita to me.
@YoYo10542
4 жыл бұрын
Are you still selling those hoodies? I couldn't seem to find them anymore...
@alejandrojimenez108
4 жыл бұрын
Right under the video, there is a merchandise section, at the end to the right there is the hoodie.
@YoYo10542
4 жыл бұрын
Alejandro Jimenez Thanks! Got one. They seemed to be missing a few days ago...
@TalathRhunen
4 жыл бұрын
what a tease ...
@hunterkohler3697
3 жыл бұрын
Well.. I start undergrad in pure math next year. Hope we don't see this for a few years.😥
@randomtiling4260
4 жыл бұрын
Is there some degenerate case where the middle is legitimately a Poisson algebra, and the whole vertex algebra is (relatively) easy to describe? Or is it fundamentally hard to construct more or less any example, even an uninteresting one. (Great video by the way!)
@MichaelPennMath
4 жыл бұрын
There are some fairly simple examples, but unfortunately the middle is never a Poisson algebra. In the case of a commutative vertex algebra all of the non-negative products are zero, and you essentially have what is known as a commutative-associative algebra with derivation. This is also closely related to a Jet Scheme in algebraic geometry. The simplest non-commutative vertex algebra is probably the Heisenberg VOA (the algebra of one free boson). It isn't too hard to get a handle on once the basic notions of a vertex algebra in general are understood.
@randomtiling4260
4 жыл бұрын
@@MichaelPennMath The connection to Jet Schemes seems interesting! Do you know any good resources for that, for someone who knows early grad student level of scheme theory/lie theory?
@MichaelPennMath
4 жыл бұрын
The connection to Jet Schemes is very new, like in the pre-print phase...
@noahtaul
4 жыл бұрын
Shouldn't the Jacobi identity be [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0? It should be cyclic.
@EliotPlaysMinecraft
4 жыл бұрын
Yes, he swapped [z,x] with [x,z]
@noahtaul
4 жыл бұрын
Queenkinghappy lol Michael’s such a goofball
@pikupal8996
4 жыл бұрын
Before this,I didn't know poisson brackets were related to lie algebra .
@mrl9418
4 жыл бұрын
More
@someguy3335
4 жыл бұрын
That ending caught me off guard
@johnalley8397
4 жыл бұрын
Hey Mikey, what's with that weird "g" character. I cannot find it Mathematica... or in google images.. little help?
@MichaelPennMath
4 жыл бұрын
It is\mathfrak{g}. en.wikipedia.org/wiki/Fraktur#:~:text=Fraktur%20(German%3A%20%5Bf%CA%81ak%CB%88tu%CB%90%C9%90%CC%AF%5D,typefaces%20derived%20from%20this%20hand.
@johnalley8397
4 жыл бұрын
@@MichaelPennMath mmm... I finally found it. Pretty obscure.
@TheMauror22
4 жыл бұрын
Wow I got brain-fucked, thank you. I can see how this relates to Quantum Mechanics, for example.
@littlenarwhal3914
3 жыл бұрын
This video is pretty self contained: uses tensor powers and makes me feel dumb haha
@davidhand9721
Ай бұрын
I must know more but I'm so confused
@cicik57
4 жыл бұрын
omg, the assoziative and lie algebra were fine, but with vertex algebra your brain just explodes 0o. just do not tell me next time there will be non-integer "multiplication" in vertex algebra, like u0.5v or u3.1415v to choose smooth the operator options :D
@ilyafoskin
4 жыл бұрын
Many of you may be wondering... why should we care about vertex algebra? That's a good question. *Video ends*
@JalebJay
4 жыл бұрын
How does someone first imagine these systems?
@joeybeauvais-feisthauer3137
4 жыл бұрын
Physicists encounter weird objects while trying to formalize a theory, mathematicians look at them and abstract a definition from them. Then mathematicians build a theory of those objects and the results can be used to help physicists
@therealAQ
4 жыл бұрын
Well, we didn't stop in a good place this time, or stopped at all
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