Why study Lie theory? | Lie groups, algebras, brackets #1

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Lie’s theory of continuous symmetries was originally for differential equations, but turns out to be very useful for physics because symmetries are manifest in many physical systems. This is the start of a series on Lie groups, Lie algebras, and Lie brackets.
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Sources:
History of group theory: en.wikipedia.o...
Lie’s theory for differential equations (actually developed further after Lie died):
arxiv.org/pdf/...
www.physics.dr...
www.researchga...
en.wikipedia.o...
Lie theory applications to mathematics: math.stackexch...
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[@mathemaniac]

So I decided to make a series on Lie theory! Hopefully wouldn't get too stressed about making another video series - last time I was hoping to see the end series simply because my interests simply waned by the last video in the series. However, because this topic is a little more closely related to what I will do in the future, I am hopefully a lot more interested in it. Anyway, enjoy! P.S. The next video is going to be a little "easy" for some people, but it is to let everyone on the same page, and honestly there will be some further and more specific motivation to study Lie groups in general.

[@itzakehrenberg3449]

I am quite happy that you are doing this & look forward to future videos in the series.

[@tissuepaper9962]

"too slow" is better than too fast in a KZbin lecture series, don't worry about making the lectures too easy. Those of us with shorter attention (or more advanced knowledge) have long since learned how to put the video on 2x speed. One of the problems I have had with Michael Penn's videos about Lie theory is that he assumes a lot of domain knowledge that I don't have. I'd much rather have it spelled out in excruciating detail and skip over the parts I already know.

[@landynillar]

Do you see how all this affect the algebraic language.i think that many mathematicians forgot that mathematics language attributes are primary to numerical attributes.

[@davidjohnston4240]

I like big brackets, I cannot Lie.

[@timbeaton5045]

On a completely different note, there was an American author of Science f=Fiction called Leigh Brackett. No relation!

[@ayearnfulhope]

“Numbers measure size, groups measure symmetry.” - Mark A. Armstrong I have been self-studying lie groups & algebras recently, this series will be an excellent companion to my pre-existent knowledge of this subject. Cheers & godspeed!

[@utof]

wow, what a quote!!!!

[@ayearnfulhope]

@@utof came across it in the text 'Physics from Symmetry', it has some really nice quotations.

[@mathemaniac]

Finally decided to make a series on Lie theory! Hopefully I wouldn't be too stressed out about making a video series again - that's one of the problems in my previous ones as I became too impatient in getting to the end of it.

[@GordonHugenay]

Great! I often encounter Lie theory, and I understand the definitions, but I still don't feel that I have a satisfying intuition. This series might be very helpful!

[@benburdick9834]

Excited for this series! Finally, something to demystify what it was I was doing in my classical mechanics course, lol!

[@loopylare]

I very much appreciate this initiative to teach and explain Lie Algebras. I am an engineer and this mathematics is not normally taught, yet it is very useful in my field (e.g. finite rotations, nonlinear manifolds, etc.). I am self taught, but the lack of formal education has made it tricky to master it. It's nice to see your videos on this and start to understand the root of the concepts.

[@Null_Simplex]

I love symmetries (groups) and I love smooth manifolds, yet I’ve never learned Lie groups despite knowing they were important to me. Thanks for this series!

[@namannarang2721]

I'm super excited this means a lot since lie groups have always been a a future plan of mine to stufy

[@carloselfrancos7205]

Can't wait for all of those videos !!

[@MaxxTosh]

I’m so excited, I’ve been waiting for a good explainer for Like theory!

[@rdbury507]

Thanks for this. I've studied Lie algebras, but the textbooks I've read usually just start with axioms for the bracket operator with little or no motivation. No one ever said "it's meant to be like Galois theory." Btw, whenever I hear the phrase "Lie bracket" I think of the sci-fi writer Leigh Brackett. I'm pretty sure they're not connected though.

[@mathemaniac]

The content is nothing like Galois theory, to be honest. However, that was what prompted Lie to study symmetries, which is something I did not know before researching for this video series!

[@LemoUtan]

Is Leigh Brackett discreet though?

[@sulfursw9286]

Oh wow, always wanted to understand this topic! I wish you inspiration and success on this project.

[@sekirokratos3755]

Can’t wait for the lessons. Keep up the good work, @mathemaniac!

[@gurkiratsingh7tha993]

Noicce content, this guy deserves 1 million likes

[@cryptonative]

Did my part

[@agrajyadav2951]

100 likes from real maths students > 1 million random yt likes

[@ANONAAAAAAAAA]

I'm really looking forward to this series! My current extremely poor understandings of Lie theory are as follows, which I really want to upgrade from. - Dealing with continuous groups means you have to think of infinitely small change of the group elements, just like deriving velocities from positional changes - For studying infinitely small changes of continuous groups, you have to investigate the tangent space spanned on the identity element of the group or the changes form the identity. - You only need to care about changes from the identity because any change can be transformed to the change originated from the identity element just like this: a '+' da = a(e '+' a^-1da) = a(I '+' da') - Plain tangent spaces are actually not enough to capture the noncommutativity of the groups, for which we need special operator on the tangent space called Lie bracket.

[@zhuolovesmath7483]

Wonderful, wonderful. I haven't taken lie algebras yet. Looking forward to this series!

[@simbarashebepete407]

Thanks for providing context. It makes the subject matter more accessible. Looking forward to the video series.

[@JakubS]

Excited for the series!

[@friendly_hologram9597]

Looking forward to this! I've come across quantum groups in some work I'm doing at the moment, but I haven't ever studied lie groups/algebras in their own right. I'll be teaching myself this topic over the summer so it's a nice coincidence that you're planning on covering it, hopefully it'll help build my intuition :)

[@morphismmap4617]

Eagerly waiting for this series, hope you could shed some light on its application in control system.

[@Achrononmaster]

@3:54 little tip: don't say "quantum spin". There really is no such thing. It's just rotational symmetry but of local structure in spacetime rather than global. Spin groups are classical. Quantum mechanics has nothing to do with spin per se, and quantized spin is not a thing. What is quantized is spacetime topology, which provides us with the local structure. What *_is_* a thing are the commutation relations between the generators, but they are classical too, however in classical mechanics since we can measure to arbitrary accuracy we don't employ them, but we do worry about commutativity when we cannot measure to arbitrary accuracy and so when the commutators do not vanish we get incompatible observables. All the "quantum" is in the entanglement structure (the reason why we cannot measure to arbitrary accuracy), not the spin structure. The spin structure is classical really, fermions are natural in classical GR (if you use the spin connexion, not the metric, or if you prefer, two gauge fields, one for positon gauge and one for rotation gauge), and can be well described in a real Clifford algebra of rotors which also works for QM. Spinors are just scaled rotors. In other words, spinors are not particles per se, they are instructions. But what for? Answer: Instructions for a theory of measurement (how to translate a laboratory frame onto a particle's co-moving frame).

[@kylebowles9820]

Dang this is going to be awesome! Looking forward to it

[@doubter7761]

Cant wait for the next video. Thank you for your work.

[@probcsh]

Never really did Lie group in grad school. Thank you for filling this void for me

[@Phantores]

I'm excited to see more after actually knowing the initial motivation

[@YindiOfficial]

AGAIN THANK YOU SO MUCH FOR THIS IM SO EXCITED 😭😭😭

[@chriskindler10]

this is going to be great. I am looking forward to the spin 1/2 chapter !

[@GiovannaIwishyou]

Can't wait for this 😊I'm studying Coxeter groups so it would be interesting to find out more about Lie's theory.

[@maxwibert]

I'm really excited to follow this series. I turned on my bell!

[@hansolo9892]

holy smokes I screamed for lie groups but I cried why it was only 4 mins. edit: the series is gonna be wild

[@1Anket]

You are a great teacher. Would also love to see a video on Fourier transform from you someday in the future

[@cykkm]

Can't wait!!! Hope it's going to be a really deep dive. Lee brackets are used in physics, both GR and QFT, but only as an applied tool, I don't have an intuition for them. Symmetries, of course, all over. I was going to learn this topic for a long time! It's so fortunate that you're starting this series, thank you!!!

[@johnchessant3012]

Looking forward to this!

[@nucreation4484]

Really looking forward to this series. Subscribing

[@ominollo]

Great intro! I can’t wait! Thanks 🙏

[@jake-o3843]

damn, i finally find a captivating series on lie theory, want to check out the other videos, only to realize it was uploaded 8 hours ago :(

[@mathemaniac]

Please be patient :)

[@mistertheguy3073]

I’m so excited!

[@rafaellisboa8493]

YESSSSSS LETSGOOOO LIE SERIES

[@MarckUrcia07]

I'm waiting for the videos.🤩

[@yash1152]

2:39 thanks for quotng some, _any_ source other than wikipedia.

[@ReaperUnreal]

Whelp, I'm excited.

[@SilentAdventurer]

It would be interesting to describe linear regression in Lie algebra terms too. I have not seen that happen before.

[@narfwhals7843]

Gaolis... what a guy. French Revolutionary. Arrested and locked up. Revolutionized math while in prison. Rejected by Poisson as incomprehensible. Gets released. Fights pointless duel. Dies at 20... If that is all i learned because of this video I'll count it a success. Looking forward to this series :) You talked about continuous symmetries without mentioning Noether's Theorem? How can you be so brave?!

[@Utesfan100]

I am hoping for some exceptional vidoes in the future. :)

[@guyarbel2387]

Thank you, can’t wait

[@alexweinberger8925]

Very excited 😮

[@CristhianDebarros]

It is good to understand The classes of symmetries, and how they are connected with a pure and simple Lie algebra. I can understand the formal definition of a space of symmetry as a finite Lie-algebra $f_{1}$ in $\{f_{1}, p\}$ since $p= n$ (so p is any integer ) , which acts freely on the functions $f_{1}$ . Then it is $\{f_{1}, p\}= Sym_{\bullet}$ since the associative algebra $f_{1}+ p|_{\varphi}$ limits all finite and simple Lie algebras. Now there exists a $\{f_{1}, p_{k}\} o that only limits to $p_{k}$ , here the Lie-algebras are semisimple in $\Psi{} (r)$ , since the equivalences in $p_{k}$ or their symmetries are only local, for example $p_{k}= R^{n+k}$ which constructs only local symmetric spaces of a Lie semialgebra, in $Sym (p_{k }):= R^{n+k}\times R^{n-1}$ .note that when a Lie-algebra $f_{1}$ is simple throughout $p$ it acts freely on $p$ and arises a concept in algebra and geometry called -Global space or group $G_{2} (X)$ Which proves that $p\in{} 1,2\} in Every semialgebra $p\in{} \{1, 2_{ 0}\} such that the algebra is only associative on $1$ . Here arises an idea from Clliford of the vector-Mukai $v$ that is semi-orthogonal in $v^{+}\to{} M$ here the semi-simple Lie-algebra of $p_{k} is studied on that vector $v$ $.... In general, a symmetry is always a Lie-algebra that is associative,

[@TranquilSeaOfMath]

Nice introduction to the topic.

[@cocoabutter5888]

These videos are incredible! What textbook would you recommend with this?

[@JCisHere778]

Lie Groups are quite the rabbit hole to go down :). But I have to disagree with the statement that Lie-Theory applied to differential equations did not catch on... I believe his reduction algorithm for ODEs is implemented in most CAS software and covers most standard methods. (Except for Integration multipliers. Which, however, are closely related)

[@PatientDeoMurhulaBuhendwa]

Nice motivation 👍

[@eguineldo]

For anyone interested in the history of mathematics, Isaak Yaglom has an amazing book about Felix Klein and Sophus Lie.

[@darkside3ng]

Amazing 🎉🎉🎉

[@g3452sgp]

What is the major outcome from Lie theory on the differential equation ?

[@user-vo3dc6je2g]

That's great! Have you charted a tentative timeline for the videos?

[@kquat7899]

Excellent!

[@richardneifeld7797]

Very informative!

[@xxABRAHAMxx]

amazing job!

[@Matt-yu8xc]

15 links in the description! Nice. How about 16th link. A link to Part 2?

[@PerMortensen]

What sort of posting schedule can we expect from this series?

[@mathemaniac]

Will not ever commit to a posting schedule. That forms a big part of the stress in my previous video series that I do not want to repeat.

[@PerMortensen]

@@mathemaniac That's fair

[@444haluk]

OH MY GOD, PLEASE DO, I WAS ABOUT TO LEARN THIS 1 YEAR LATER IN UNİVERSİTY, YOURS WİLL BE FAAR BETTER

[@edbertkwesi4931]

I LIKE DIFFERENTIAL EQUATIONS A LOT LETS GO

[@mMaximus56789]

I'm working rn on optimization on Lie Groups, would be nice to see how to apply functions (specifically non linear) to a group

[@landynillar]

In the realm of mathematical exploration, a visionary mathematician named Eli discovers a hidden dimension within Lie algebra-a realm where abstract symmetries take shape. As Eli delves into this uncharted territory, a new kind of geometry begins to emerge, challenging conventional notions. In this geometric landscape shaped by Lie algebra, memorizing intricate structures becomes the key to unlocking its secrets. Eli's mind becomes a repository of complex ideas, where the memorization of Lie group transformations and infinitesimal elements becomes a poetic dance of understanding. As Eli navigates through this geometric tapestry, the traditional notions of points, lines, and surfaces blur into a symphony of interwoven concepts. Memorizing the subtle interplay of Lie algebraic elements transforms Eli into a custodian of an otherworldly geometry, where the language of symmetries dictates the rules. The entasis of this mathematical odyssey lies not only in the intricate memorization of these complex ideas but in the revelation that this new geometry offers a profound glimpse into the nature of the mathematical universe-a journey where memorizing becomes a profound act of communion with the intrinsic beauty of abstract structures.

[@landynillar]

Well this nonsense tale maybe not so crazy. Lie theory it will have mayor impact in mathematics, at the language level

[@AymanFakri-ou8ro]

awesome, thanks!

[@watcher8582]

I'm not sure if "...still very useful, not necessarily differential equations" is accurate. Especially if you then pull up their use in physical systems, which are for a large part of it part all governed exactly by differential equations. The symmetries of these equations and Lagrangians (be it in classical mechanics or field and particle theory) are how they come into play. Maybe it can be said that they did "not dominate the study of differential equations" (as there are many methods in that field), but in the other direction when Lie groups are applied, then it's exactly because of their relation to differential equations. Even if in the math department the Prof's there will teach these groups in abstract isolation (just how they present group- and manifold theory also).

[@mathemaniac]

Maybe I should have said the use has gone well beyond DEs. I still stand by "did not dominate the study of DE", though.

[@watcher8582]

@@mathemaniac I'm just nitpickin'

[@g3452sgp]

This is a sort of interesting!!

[@khadijaaithmid5536]

Thank you

[@hashdankhog8578]

Why isn't continuous symmetry used for differential equations

[@frankjohnson123]

I think you will address this with the prereq video, but I'll ask in case. I think Lie theory is more relevant to me than Galois theory, but I was wondering is Galois theory recommended to study first?

[@mathemaniac]

No, not at all. Galois theory served as a motivation for Lie, but almost no physicist, perhaps except from me, need to know Galois theory, but do need to know Lie theory.

[@frankjohnson123]

@@mathemaniac much appreciated!

[@george56834]

I thought this video was going to be about something else. Like how to find out if someone is lying or not. Does anyone have a theory about that?

[@陳力歐-d2d]

please keep going～

[@Alan-zf2tt]

Rubbing hands with glee!

[@asmithgames5926]

My hips don't Lie.

[@mathemaniac]

And I'm starting to feel it's right :)

[@KevinBalch-dt8ot]

Therein LIEs the question!

[@Michael-cg7yz]

oh god the subtitles

[@ClabClab]

commenting for the algorithm

[@bhayasur]

See you soon, But now I gonna read all about it .

[@landynillar]

Ok just with the first 3 minutes of listening to this

[@ucngominh3354]

hi

[@BoyNextDoor1729]

first

[@Kelvin-ed6ce]

Lie lied about all these.