The universal enveloping algebra
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- Опубликовано: 6 сен 2024
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Definitely agree that these videos are more valuable than contest problems videos. As with the other advanced videos, I would appreciate some motivation or applications. It doesn't have to work out properly, but I would want to have an idea what are these objects good for.
Completely agree
Lie groups are important in particle physics, and this universal enveloping algebra is related to symmetries and operations with these same Lie groups (disclaimer: it's not my specialty)
@@pseudolullus you can construct the different representations of the Lorentz group (or it's universal covering group as Weinberg said) and study how this reflects the properties of different kinds of particles. It's a wild trip, but certainly a interesting one.
Glad to hear you won a math contest and are developing a predilection for more abstract math.
Fundamental particles can be defined as unitary representations of the Poincare (Lie) group.
The Lie algebra describes the infinitesimal generators of a continuous symmetry (Lie group), and are related to the operators of quantum mechanics.
There's definitely UVAs lurking in there somewhere...
100% want to see the representation theory stuff!! keep it up Dr. Penn!!
I would love to see an equally long video showing how these objects are being used in the wild or where they appear, or even better how they were used to solve a problem. Like for example with generating functions: when I first learned about them I thought "OK, cool we have a fancy way of bundling up sequences into series, what now?" and when I saw how cleverly they are used to come up with all kinds of information about the sequence itself, this totally blew my mind. I'm curious what this crazy thing is being used for :)
i completely agree...my mathematical background relates to astrophysics so when i see math, i immediately want an understanding of what the point is...
he's done vids on quantum mechanics math that were amazingly enjoyable...
Pure mathematicians are not burdened by the “what now?” question. In fact some of them worry that the precious offspring of their brains will be tainted by vulgar “real world” applications.
@@MyOneFiftiethOfADollar that's why I referred to seeing how the math is being used to "solve a problem", which could be a purely mathematical problem :)
Imagine have to lean a mathematics with no purpose what so ever. And, You learned. Now justify the wasted space of knowing it. Bent forks are unique but the are neither valuable nor useful. Check your applied nihilism and explain when to use this algebra and when it is a waste of time. It obviously does not envelop universes as it promised.
@@axiomfiremind8431 you have only revealed that you are unaware of any utility or value a bent fork may have.
definitely like the rep theory content! would be cool to see a follow up maybe talking about PBW theorem and associated graded algebras to continue the discussion of degree
Always watch Dr. Penn to cure any idea I have about being "smart." I once asked my high school algebra teacher if algebra was the hardest math invented. I remember that he smiled and said, "no."
Well, the question and answer are a bit subjective. I still find nonlinear partial differential equations to be the most difficult. At least algebra has rules. I'm not sure that we have even scratched the surface of nonlinear PDEs.
And when you do solve a PDE, let them act over theses algebraic structures. . . .
YESSS ADVANCED MATH DAY!!! I love when you post videos closer to your area of research!!
I love the more abstract and in depth content !
Sleeping with SU(3)xSU(2)xU(1) is the best experiment I've ever had!
Standard model of particle physics, right?
@@MrFtriana Right (:
Absolutely love when you do these condensed lectures on some more abstract topics. It's a bit of a gamble from you but they're the videos I always watch immediately
Yes, but there do seem to be more commercials needed to pay for them. I don't like commercials, but I can't stop watching Dr Penn.
A series on Geometric Algebra, much like the Abstract Linear Algebra series, would be much appreciated.
Please, absolutely, do more of these! I first subscribed because of your videos on VOAs; I don't even watch the contest videos, and I've probably lost a bunch of these sauced ones in the noise, but there's a noticeable lack of good videos explaining nonstandard topics at and beyond undergraduate levels!
Most of what I've found, even for things as tame as multilinear algebra, including those in print, aren't quite able to motivate things as well as you.
I'd love to see a comparison of this to Clifford algebra.
I did notice while watching this that it appeared _very_ similar to Clifford Algebra. The part about quotienting by looked familiar, and ignoring the Jacobi identity, that's basically the Geometric Algebra definition of the outer product of vectors, but with an extra factor of 2. Using the commutator product instead of the outer product, it actually satisfies the Jacobi identity. I think the only real difference is that Clifford Algebra is instead built from an inner product (v^2 is a scalar), where as the algebra given here is built from the lie bracket, which plays a similar role as the outer product.
I guess they're kind of mirror images of each other in that way. Given that geometric transformations are inherently associative, it makes sense why the _Geometric_ Algebra would derive from the version with associativity rather than the Jacobi identity.
I too was thinking about Clifford and Cayley-Dickson algebras
@@angelmendez-rivera351 Clifford Algebra often tends to abuse the notation and just call the quadratic form the "inner product." The fact that it's not positive definite is honestly a feature rather than a bug, especially when dealing with non-euclidean geometries like Minkowski spacetime.
Actually Clifford Algebra abuses the notation even _more_ and calls the inner product the lowest possible grade output between the grades of the two inputs, which isn't even always symmetric. For example, the "inner product" between a vector and a bivector is ANTIsymmetric, and returns a vector rather than a scalar.
Most formal definitions tend to use the quotient as you described, but most introductions don't actually mention the quotient at all. Instead they simply assert the existence the existence of an associative bilinear product between vectors where a vector times itself is a scalar. They then show how that one statement is enough to generate the entire algebra and determine its properties.
I strongly suspect that this is related to what makes Clifford Algebra attractive to many people in the first place. I'm sure I'm not the only person who uses it who finds tensors and their construction to be too abstract, and so general that it could describe almost anything. For these people, like myself, we're more likely to call it "Geometric Algebra," and treat the algebra as little more than a way to describe geometry and geometric transformations algebraically.
For example: `bavab` I don't actually know what b, a, and v are (though v is probably a vector), but what I _can_ say is that it's a reflection (either across or along) a, followed by a reflection (either across or along) b, and together they form a rotation of some kind. That may be a translation, or even scaling, but I already know for certain that `ba` is a transformation object being used to transform v.
@@angelmendez-rivera351 *The construction of the Clifford algebra is a deformation of the construction of the exterior algebra, while the universal enveloping algebra is a deformation of the construction of the symmetric algebra. Remember, the exterior algebra Λ(V) := T(V)/, where T(V) is the tensor algebra, the Clifford algebra Cl(V, Q) := T(V)/, the symmetric algebra S(V) := T(V)/, and the universal enveloping algebra U(V, [•, •]) := T(V)/
Why do people like adding and multiplying integers? It has a nice ring to it.
Comathematicians are devices for converting cotheorems into fee.
This is crazily interesting and cool, please keep making these kind of videos. I woule also be totally psyched for a video on representation theory!
Would be great to also see how this relates to vector fields on S^1 and more generally to left-invariant differential operators!
I absolutely love videos like this. Thank you Prof Penn.
You should do a collab with a physicist about these ideas! But also definitely do more of these videos. I’d love some more stuff on vertex operator algebras!
A rep theory playlist would be insanely cool
Really liked this one! Would love to see a continuation of this video
How about a video (or several) about tensor products over arbitrary rings? Maybe just commutative rings.
Representation theory will be a great addition
I'd definitely like to see more on this sort of stuff. I'm curious if there's perhaps some sort of basis for a universal enveloping algebra, where the elements have some sort of non-negative weight associated to them, similar to degree, so that transformations using the commutation relations preserves this weight.
I would like to see more "survey" videos: videos with a general topic that discuss in a shallow way many other topics. Perhaps talk about rep theory and list briefly a dozen results in the field, then fill it with links to videos that go into more depth (or ask your audience which follow-up videos should have the mandate!)
These types of videos are the bestest! It would be nice to see some differential geometry o tensor calculus, at least an intro
Would love to see some content about representation theory!
I love your videos, especially ones like these. The only frustrating thing is that I’m always left wanting more because we’ve just scratched the surface of something
Michael: does a vid on abstract maths with no mathematical mistakes
Also Michael: h comes before e and f in the alphabet
Anyway I'd totally like something on representation theory, yes please!
Yeah can you please make a video of representation theory? I never studied it in college.
Yeah I think this type of somewhat advanced content is more interesting than your usual contest video. Keep it up, maybe you could add some motivations/applications for the topic at hand
I like these types the most.
I just love it when you say things like that! Representation theory is a fascinating area. Your take would be extra interesting because of the groundwork you laid in videos like this.
Very interesting. I'm glad I don't have homework to turn in to be graded on this video. But these structures are useful in physics. I see them in some unified field theories. I'm not sure that I could find a better explanation than Dr Penn gives. Thanks.
I would love to see more videos on algebra, your explanations are great and i can't really find a lot of alternative content on these topics on youtube
Great stuff. More, please.
Recognizing so much of this from quantum mechanics class! h e and f are basically the pauli spin matrices, not to mention commutators on matrices
yeeees! I think it would be great for this channel to start doing courses in this topics, just like "The Bright Side of Mathematics" if you have ever seen him. I think a good course with such a great instructor is really hard to find even at the university.
I'm a big fan of these kinds of videos because they show a side of math that isn't very commonly found on RUclips
These kind of videos are what makes your channel unique in my opinion, would love to see more of this type of content.
I’ve thoroughly enjoyed all the videos on more advanced topics, especially the VOA series. Those saved me a lot of time I would’ve otherwise spent being unproductively confused trying to step into the subject. I’ll be eagerly awaiting whatever you have planned to follow this one!
Surprisingly easy to follow given how abstract this is gets. Thx!
Very interesting video/lecture, thank you! I jumped right in, with little prior knowledge, and I understood basically everything. I'll be watching or reading something more on sl2 and Witt algebras, because they look interesting and it's my first time seeing them.
@0:18 "'Universal enveloping algebra' just has a really nice ring to it"
No pun intended
28:59
Please Micheal, a full playlist of Lie Theory, just like the ones you’ve done for Ring Theory, would be awesome! ❤
@ 7:20 has representation theory video been completed yet?
And, as part of wish list, regarding algebraic geometry, differential geometry, other close relatives of these but with certain twists to fit other things (topology?) is it possible, doable to do a certain base explanation of which Michael presents and the ta-ta-taaaan other speakers, guests, learned colleagues, other RUclips channel hosts .... with experience in those other geometries - probability theories? too - do follow up videos where these experts introduce where their topic converges and/or diverges and/or somehow differs from Michael's algebraic base case?
Of course it would be fun for channel watchers but it would be superb fun if Michael and guests had fun doing it as well.
I mean, math is serious but it is also good fun as well - I think the academic side of things hammers out fun and beauty aspects (partly due to curriculums and learning outcomes rules and so forth)
C'mon Michael - if anyone can pull it off you can ⭐
I really enjoy those more advanced and longer videos.
Competition problems and super convoluted integrals are not my thing tbh.
I was trying to pick some bits from Algebraic Geometry a while back and the last section reminded me of manipulating basis vectors, which I think makes sense because AG is an application of Lie Algebras IIRC.
Super cool video!
These are my favorite kind of videos for sure! I would love a simple and thorough explanation of Lie algebras, where they come from, how to think about them, etc. I have no idea what topic they are a part of, and my attempts to figure out what they are on my own haven't worked so far!
Also if there's more to know about this "universal algebraic construction" technique. You did the same thing in the tensors video and I'd love to hear about why you use this technique instead of the axiomatic technique I see in a lot of other definitions. Although that might be more of a philosophical topic? I don't know.
At 13:45 you gave an example of how to identify the tensor product of two vectors with matrices, using what is essentially the Kronecker product of the first vector and the transpose of the second one. My question is if you have three spaces U,V,W (e.g. C^{n_1}, C^{n_2}, C^{n_3}), whether you would identify this with a three-dimensional array instead of size n_1\times n_2 \times n_3, or whether you would take the Kronecker product of the first vector times the second vector transposed, times the third vector, which again yields a matrix?
To answer your question, I'd really appreciate more videos like this one!!
Great work!
Super cool video! I would really like to see representation theory as well.
Can we get a couple of videos on the semi direct products of groups and the schur-zassenhaus theorem so we can classify all groups of order 2022 before the year's end? 🎉
I would like to see more of these.
I am begging for someone to please ackowledge that this man occasionally says wild stuff about being from the future or being a vampire.
Love the shirt Michael, the bright colours look good on you 👍 and a fascinating lesson as always!
I'd love to hear your take on the projective and conformal geometric algebras that folks like Steven de Kennick and Anthony Lasenby have been pushing. The computer graphics and robot kinematics people are all over them.
I love this type of videos ❤ I find them fascinating
I love this videos please do more of this!
More of these videos please!
These are my favorite videos, I would definitely like to see more
Would love the representation theory vid
I love these types of videos!! would love to see a little bit of the history of development of concepts like this. what problem were they created to solve? has this concept unlocked something interesting in another part of math?
Very nice and interesting video. Would be a good idea to go on along this path.
While the Integral problems are a joy to watch (i personally dont really like the math contest vids), this video and the previous vids on tensors/sl2 are just on another level and highly informative and interesting. While the amount of work that this type of video takes is probably a lot higher than integrals/contest problems the payoff is definitely worth it. Thank you for making these.
And as someone studying physics, it would be extra exciting if you made a video on f.e. Operator algebras
I would love to see more representation theory stuff, I definitely prefer the more theoretical videos
I'd love to see a video on chain complexes or similar
I love this video! I don't mind some spicy representation theory
I'm 11 months late, but so it goes. I'd love to see a construction of the Temperley-Lieb algebra. I come at it from a knot theory perspective. Cheers.
I had some trouble following along, but it was definitely an interesting video. And I would totally be interested in a video on representation theory.
As usual thank you for the video.
Yayyy I was waiting for a video like that!!
This is great, thanks. Would love to see that representation theory vid!
Loved that. Thanks.
These videos are great! I would love to see more!
12:49 Can you explain why you can say V tensor W is "essentially the same as" M_(2x3)? I see how it can be similar but from the representation, it looks like V tensor W has 5 degrees of freedom, alpha beta, a b, and c, while M_(2x3) has 6, one for each matrix component.
There are only 5 degrees of freedom in v tensor w, but the Span in the definition gives you more. Not all of the linear combinations of terms simplify to a single term.
An element of V⊗W is called a simple tensor if it can be written as v⊗w, but most elements of V⊗W are linear combinations of simple tensors which cannot be factored. This adds the 6th degree of freedom.
Another way to look at it is that with a simple tensors, one of the matrix elements can be calculated from the other 5.
Great content, highly useful!
This sort of videos is my favourite! :D
Please post more about this universal enveloping algebra guy. :D
Is there an algebraic construction (similar to Cantor's proof of the bijection between 𝔸 and ℕ) which constructs not only polynomials, but also more complicated arithmetic sequences: like those involving tetrations, super roots, super logs, then pentation and its and inverses, during each new height of the construction?
I would very much enjoy learning about representation theory :)
Another question: it's always possible to write a polynomial in U(g) as a monomial in U(g) using the relations xy-yx =[x,y] ?
"Very nice ring to it" Budum tsssss
A little off topic, but this strange Lie letter g comes from the german Kurrent alphabet, in 1911 it was modified to a slightly simpler Sütterlin and then forbidden by the nazis in 1941.
Excellent video. I look forward to more of them. One point though. At about 22:20 when you start simplifying your monomial example, you seem to be assuming the associative law for multiplication in the universal enveloping algebra. Of course, it does hold, but you probably need to prove it as it doesn't follow trivially from your definitions.
Multiplication in the universal enveloping algebra is just the tensor product, which is associative. In fact the universal enveloping algebra is constructed in the first place because we want some associative algebra that has analogous structure to g, which is most certainly not associative.
This is awesome! Please more!
This was quite clear. Thanks for making something that seems off limits for the non-mathematician at least somewhat accessible. If you're taking votes, Hopf Algebras, please!
The comments about charge conservation came out of the blue. Would love to know more about how it is defined and why it is conserved.
More videos like this please!
@MichaelPennMath 0:55 I think LaTeX is pronounced “lay-tek”, at least that’s what it said in the LaTeX book I’ve had since the 90s.
Commenting for the yt algorithm, cool video keep it up
Yes! Please make a video on representation theory.
7:16 yes i definitely want it!
Also, how about Clausen Integral values of alpha*pi and how it's related to sigma-over-k of (nk+a)^-2?
Yes, I would like to see a video on representation theory.
15:45 This reminds me of Fock space.
yes, I'd like to see something on representation theory
At 16:42 it should be T(g), not T(V)
eefeh definitelly describes my emotions here
I’d like to see the representation theory video!
7:10 Yes please do a video on rep theory
7.22 yeah i looking for the video