the derivative is a LIE

Рет қаралды 29,392

Michael Penn

Күн бұрын

Пікірлер: 80

@ddognine Жыл бұрын

21:10 Minor correction: should be partial wrt y, not x, which allows it to double up as 2e.

@gennaroponsiglione1098 Жыл бұрын

Same doubt here

@SuperDreamliner787 Жыл бұрын

This reminds me of my quantum mechanics classes a couple of years ago. Now I am teaching how to add fractions. 😂

@MH-sf6jz 8 ай бұрын

Abstract algebra is a tough course to teach

@chester_m Жыл бұрын

Michael I extended the playlists on my channel of videos on your channel a bit. In particular there is a playlist with over 90 of your videos that are mentioned in the OEIS. If you like they can be on your channel. I can also send you the list I use to make the playlists. Free of charge, just as a thank you for your work.

@martin13423 Жыл бұрын

I really enjoy these more exploratory and higher level videos, they're really interesting !

@theartisticactuary Жыл бұрын

Never has the line separating pure maths from quantum mechanics looked thinner.

@giannisniper96 Жыл бұрын

super excited about a video on the representations of sl_2(C) 😄

@benjiusofficial Жыл бұрын

Adding another voice to the choir

@schweinmachtbree1013 Жыл бұрын

the Penn fact at 7:00 could have done with a comma - it might be confused as "componentwise [addition and multiplication] ..." but what is meant is "[componentwise addition], and multiplication ...". (also it uses an epsilon ε instead of a membership symbol ϵ)

@MikeGranby Жыл бұрын

It could also have some with a graphic that looked vaguely like Penn..

@jordanraddick505 Жыл бұрын

Damn you, Penn, your clickbait worked on me! *shakes fist* Edit: Damn you, Stephanie!

@MichaelPennMath Жыл бұрын

Under no circumstances am I missing this opportunity. -Stephanie MP Editor

@paulkohl9267 Жыл бұрын

Hey MP, loving the video editing improvements. Excellent. 🙂

@sinecurve9999 Жыл бұрын

Top tier mathematical themed dad joke.

@enisten Жыл бұрын

Click-bait in disguise of a pun (though, I stayed because of the content)

@georrgy Ай бұрын

These series are perfect! You are the first who gave me some naive idea what Lie algebras are, so now I am mentally ready for the appropriate course. Thank You very much 👍

@VOLKOV9 Жыл бұрын

came here to applaud your change away from the clickbait title. a lot fo math/science youtuber are heading the other way. I get why they do that, but I want to celebrate you going this way.

@dodgsonlluis Жыл бұрын

I got stuck when the calculation leads to z^(m+n-3) and not z^(m+n-1) as needed for L(m+n). If we set in the definition of L(n) z^(n+1) instead of z^(n-1) then works fine.

@krisbrandenberger544 Жыл бұрын

Actually, shouldn't the commutator of L(m) with L(n) be equal to (m-n)*L(m+n-2)?

@dodgsonlluis Жыл бұрын

@@krisbrandenberger544 Sorry, the definition corrected leads to z^(m+n+1), as needed for L(m+n).

@dodgsonlluis Жыл бұрын

@@krisbrandenberger544 Check his video on the topic. kzbin.info/www/bejne/Z36uhaZ-e6psl6M

@krisbrandenberger544 Жыл бұрын

@@dodgsonlluis Perfect! Thanks!

@00-102 Жыл бұрын

In the example 2，why there is an alpha(0), it's in span{}; and how can we get m in apan{}?

@dmytrolevin738 Жыл бұрын

5:40 But if [y, z] = yz, then D(y) = [x, y] = xy, and D([y, x]) = D(y)z + yD(z) = xyz + yxz = 2xyz (if the "multiplication" is a usual multiplication, commutative and associative). What is meant exactly by D and that "multiplication" here?

@JohannPetrak Жыл бұрын

Best clickbaity title in the history of KZbin.

@lexinwonderland5741 Жыл бұрын

This is always my favorite video content! Thanks, professor! Can't wait for more about Lie algebras and VOAs -- i'm always thrilled to hear about your research, but even educational videos about things like representation theory make my entire day!!

@Wielorybkek Жыл бұрын

video about representations of sl2 or we riot

@pizzamidhead2183 Жыл бұрын

22:12 good place to stop

@MrFtriana Жыл бұрын

Ah yes, conmutators and derivatives, one topic frequently found in physics in quantum mechanics. It's a awesome topic that mix algebra and calculus.

@ultrametric9317 7 ай бұрын

I think I will go through your entire course - do you get to the part where translation groups come in? I never had a course in Lie theory and my memory is foggy.

@tubepkn Жыл бұрын

I like how he spells Leibnitz's name: LIEbnitz

@bjornfeuerbacher5514 Жыл бұрын

Actually, the _real_ spelling is Leibniz. He even made _two_ typos. :/

@reinerwilhelms-tricarico344 6 ай бұрын

In your Heisenberg Lie algebra starting at 11:05 I can't see how this jives with the more commonly known Heisenberg-algebra or group. I don't recognize any of it.

@cmilkau Жыл бұрын

When I first learned about the derivative, I always felt unhappy about it. I wanted something that I would now call tangential space (on the graph of a function). I guess it made sense at the time of Newton and Leibniz, but with the advent of special relativity, I again get the feeling a tangential space is the better abstraction.

@davidgo3759 Жыл бұрын

Hello Michael. I really love all your videos. In this one there is something that I don’t get. Is D: J->J / D(v)=[x,v], for any alternating bilinear map [ , ] satisfying the Jacobi identity, a well defined map? What is up with the other vector ‘x’? Besides, I think you cannot go beyond the expression D([y,z])=[D(y),z]+[y,D(z)] since when you take later [a,b]=a·b, firstly [ , ] is not anticommutative, so [ , ] is no longer alternating, and secondly, by the previous definition of D, now becomes D(y)=x·y(x) automatically (we have to assume at that point that J is the vector space of real functions and ‘y’ is a function of the real variable ‘x’) and there is not a derivative operator anywhere that allows you to get to the expresion D(y·z)=D(y)·z+y·D(z).... can you explain me where I’m wrong? Thank you.

@AJ-et3vf Жыл бұрын

Awesome video. Thank you

@tomholroyd7519 Жыл бұрын

LOL yeah it's Leibniz. In German, when you have "ie" or "ei", it's pronounced like the second letter. "Lei" is "lye". "Lie" is "Lee"

@Toranx Жыл бұрын

Actually it's "Leibnitz".

@Etothe2iPi Жыл бұрын

Liebnitz instead of Leibniz, ROTFLMAO. At least, there's no magic in this video, when something miraculously changes on the blackboard.

@MichaelPennMath Жыл бұрын

There's magic in every video. Math is magical. :) -Stephanie MP Editor

@jossarian Жыл бұрын

A general Lie Algebra may be represented(!) by the set of nxn matrices over a field, here gl(n). Further, certain restrictions like trace zero, upper triangel, skew symmetric and many more .... , these subclasses of nxn matrices represent also Lie Algebras. Next, these Lie-Algebras may act as endomorphisms of some n-tuple vectorspaces.

@mtaur4113 9 ай бұрын

Title is straight out of the 19th century math flame wars.

@mokouf3 Жыл бұрын

When you see "lie" in mathematics...

@edwardlulofs444 Жыл бұрын

Good one, thanks.

@APaleDot Жыл бұрын

I've always wondered why the cross product reminded me so much of the product rule. It all comes back to that damned Jacobi Identity! I would love a video on some sort of intuition about the Jacobi Identity. It seems to show up a lot when discussing rotations, but what the hell does it have to do with rotations? And what the hell do rotations have to do with the product rule?

@MrFtriana Жыл бұрын

Maybe it is because if you work with finite rotations you don't get the same result when you change the order of the rotations. At least i think that this is a plausible explanation.

@APaleDot Жыл бұрын

@@MrFtriana Not sure what you mean by this.

@joshuagrumski7459 Жыл бұрын

Well, I don’t know toooooo much, but Lie Groups are groups that are manifolds as well, and Lie Algebras, from what I know, are the tangent spaces of the Lie groups. SO3, the group of rotations, has a corresponding Lie algebra, so3 (lowercase), and so3’s standard basis vectors obeys a Lie bracket relation using the standard commutator. So, from my understanding, the reason why there is such a big connection is because Lie algebras tend to be tangent spaces to a Lie group, which means that the algebras tend to be highly related to derivatives in some sense. As for cross products, I believe that (R3,x) is isomorphic to so3, and so the two are highly related in that sense? But idk for sure, others may add/correct me

@tw5718 Жыл бұрын

Just to clarify, is this the video formerly known as Derivatives vs Lie groups, 2 sides of the same thing? Added this to watch later, and now its different, I think.

@tw5718 Жыл бұрын

After watching it is. First off, interesting video. I would like to provude my 2 cents about the name change. I think this title might be better for getting clicks in general. However, I'm not sure that many of those extra clicks will necessarily be your target audience. If I was a calc 1 student I would click instantly, but not really follow any of this video. I am currently trying to work through QFT and as such Lie groups are of very high importance to me, particularly ones that offer intuition, as opposed to definitions and calculations. The old title promised insight and intuition, which is why I saved it, and the video delivered. However, had I not seen the old title and known the content of the video, it's quite likely that the pun would have gone over my head, causing me to just write it off as a click baity video, and not watch it. All said and done, I'm really glad I watched it, and maybe the pun would have occured to me, but sometimes I'm slow on those things. Take that for what it's worth, just keep giving more Lie videos (and other high quality educational content). But more Lie videos.

@MichaelPennMath Жыл бұрын

welcome to the wonderful world of A/B testing. -Stephanie MP Editor

@tw5718 Жыл бұрын

I most definitely understand it. I simply wanted to offer my take on this particular title selection. I could very well be the minority. I'm not sure what it looks like on your end, but I saved the video when it was the OG title, but didn't watch it until it had changed. Not sure if you get that statistic or not.

@JCCyC Жыл бұрын

Shamelessly clickbaity title but I just can't get mad about it. 🤣

@catbertsis Жыл бұрын

10/10 clickbait, love it

@CamEron-nj5qy Жыл бұрын

That is one strange way to write "g"

@kilianklaiber6367 Жыл бұрын

Just one question for the examples. Do these differential operators form a vector space?

@paulkohl9267 Жыл бұрын

I hope you get to q-deformable Lie algebra's one day. Did an undergrad math paper on the topic. I would really like to grock the topic better. Then one day Homotopy Type Theory (HOTT) too!