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.
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)
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.
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 👍
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?
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.
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.
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.
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.
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?
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.
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
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!!
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.
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.
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.
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.
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.
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!
This reminds me of my quantum mechanics classes a couple of years ago. Now I am teaching how to add fractions. 😂
Abstract algebra is a tough course to teach
21:10 Minor correction: should be partial wrt y, not x, which allows it to double up as 2e.
Same doubt here
Never has the line separating pure maths from quantum mechanics looked thinner.
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.
I really enjoy these more exploratory and higher level videos, they're really interesting !
Damn you, Penn, your clickbait worked on me! *shakes fist*
Edit: Damn you, Stephanie!
Under no circumstances am I missing this opportunity.
-Stephanie
MP Editor
Top tier mathematical themed dad joke.
Click-bait in disguise of a pun (though, I stayed because of the content)
super excited about a video on the representations of sl_2(C) 😄
Adding another voice to the choir
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)
It could also have some with a graphic that looked vaguely like Penn..
Hey MP, loving the video editing improvements. Excellent. 🙂
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.
Actually, shouldn't the commutator of L(m) with L(n) be equal to (m-n)*L(m+n-2)?
@@krisbrandenberger544 Sorry, the definition corrected leads to z^(m+n+1), as needed for L(m+n).
@@krisbrandenberger544 Check his video on the topic. th-cam.com/video/1MKTsHFE9aA/w-d-xo.html
@@dodgsonlluis Perfect! Thanks!
In the example 2，why there is an alpha(0), it's in span{}; and how can we get m in apan{}?
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 👍
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?
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.
video about representations of sl2 or we riot
Best clickbaity title in the history of TH-cam.
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.
22:12 good place to stop
I like how he spells Leibnitz's name: LIEbnitz
Actually, the _real_ spelling is Leibniz. He even made _two_ typos. :/
Ah yes, conmutators and derivatives, one topic frequently found in physics in quantum mechanics. It's a awesome topic that mix algebra and calculus.
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"
Actually it's "Leibnitz".
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.
Liebnitz instead of Leibniz, ROTFLMAO. At least, there's no magic in this video, when something miraculously changes on the blackboard.
There's magic in every video. Math is magical. :)
-Stephanie
MP Editor
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.
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?
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.
@@MrFtriana
Not sure what you mean by this.
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
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!!
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.
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.
welcome to the wonderful world of A/B testing.
-Stephanie
MP Editor
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.
Wait but with the example of the derivative operator and [a,b]=ab, this bilinear map isn't alternating, right? So it isn't really a lie algebra
Title is straight out of the 19th century math flame wars.
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.
When you see "lie" in mathematics...
the way you misspelled "liebnitz" could be construed as a clever pun... 6:10
Just one question for the examples. Do these differential operators form a vector space?
Awesome video. Thank you
Shamelessly clickbaity title but I just can't get mad about it. 🤣
That is one strange way to write "g"
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.
BTW, the math symbology of TeX/LaTeX is the official representation of mathematical equations and formulae by the American Mathematical Society.
Good one, thanks.
Liebnitz? Was this typo made on purpose?
Yes. That's my story and I'm sticking to it.
-Stephanie
MP Editor
10/10 clickbait, love it
The title 💀
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!
Very useful in control theory
Liebnitz rule? seems like Lie + Leibnitz, nice mix.
Leibnitz rule
Science is a lie. Sometimes.
commenting to say I want that representation video, also commenting to ask for more differential forms videos
William Rhys likes fire truck
Vroom 🚒
that is not how you draw a g bro
A real g would be a drawing of me foo
Hi
The name of the guy teaching is Michael Penn.
The solution of the golden sequence video is wrong! Read the comments!