The h Bar

12:33 AM

it makes sense, because those operators show direction of propagation of the field

by the way I read this statement in a few places that quantization is a mystery but second quantization is a functor, what does that supposed to mean?

The way I vaguely understand it, there is no natural way to quantize a classical system, but perhaps second quantization is natural????

for instance, the way they quantize a surface, they consider complex line bundles and take sections as the Hilbert space, but I have no clue why it's being done in this way.

julien surel

4:01 AM

by the way, I looked at Feynman's book. It's so neat. Does anyone know if it contains any QED in it?

JingleBells

6:09 AM

@PM2Ring I know English, Bulgarian and a bit of Russian

i am so olequently spokent on it languiges

John Rennie

8:44 AM

In other news my niece has got the grades she needed to go to Cambridge.
I just hope Cambridge is prepared for what is about to hit it.

JingleBells

9:04 AM

f college

John Rennie

@JingleBells Girton

girton.cam.ac.uk

Spade 000

In the case of a string fixed at both two ends. Supposed if I plucked it and this displacement can be modeled derministically with the function y(x,t). What is it that we hear? Is it at position y(1,t)? at y(2,t)? at y(3,t)? or the sum of all this points?

I was trying to synthesis a simple plucked string.

John Rennie

You hear the pressure waves created in the air by the string as it moves.

I don't know what the transfer function is for energy in the string transferring to energy in sound waves, but as a rough start the spectrum of the sound waves will be the same as spectrum of the vibration on the string.

i.e. if you Fourier transform the function y(x,t) to express it as a sum of sine waves then the sound waves will contain the same spectrum of sine waves.

Spade 000

9:26 AM

@JohnRennie I think I got it now. It's just a matter of imagining a sound pressure travelling at a space. Still it confuses me that a 1D string produces a pressure wave and it depends on where we are placed at to hear it.

JingleBells

@JohnRennie ?

John Rennie

@JingleBells I thought you were asking what college my niece is going to ...

JingleBells

@JohnRennie oh no, f stands for f***, but congrats to your niece! I'm sure she'll do brilliantly. Nevertheless, i'm not a big fan of universities and colleges and I should probably stop expressing that every chance I get :P

I understand why college and universities exist, but for me, it's not the right thing. But if anyone wants to become a physicist, doctor, engineer, lawyer, etc, it's required and I support it.

Spade 000

@JingleBells "it's not the right thing"? What do you mean by that?

JingleBells

@Spade000 I mean, that I don't feel like university is right for me. I do not think I should go to university and I don't want to go to university.

Spade 000

9:50 AM

I might first try using the Karplus-strong algorithm because it's more simple to implement. Some great technical report about it was math.drexel.edu/~dp399/musicmath/Karplus-Strong.html

bolbteppa

10:56 AM

@JohnRennie very good news

ronak jain

11:09 AM

Hi everyone

Can someone tell me that what is the value of torque on a electric dipole on Electric field . I know that it is P X E . But when I solve it I find it is -PESin(theta) . Am I right or not ?

ronak jain

11:25 AM

Is there anyone active in the room ?

Charlie

11:52 AM

@ronakjain cdn.educba.com/academy/wp-content/uploads/2019/08/…

JingleBells

12:12 PM

I was wondering, in the case of capitalism, there are rich and poor. Rich people have created/contributed value to society and that's why they have money. Their wealth is directly or indirectly a reflection of how much value they have created for people. That I think is fair. What troubles me a bit is wealth inheritance. No one should be blamed for being born in a rich or a poor family. The thing is, why not ban inherited wealth and make everyone, after becoming 18 or whatever, start with x

amount of money? (It's a stupid idea and I have not thought about it deeply)

Charlie

What happens to someone's wealth when they die under your system?

JingleBells

It gets distributed to society (i.e it can go to the money people get when they turn 18, say 10k or something)

Charlie

Wealthy people and by extension the companies they own have enormous influence over politicians.

If you can't get those people to agree to a slightly higher minimum wage and expanded worker's rights what makes you think you can convince them to distribute their entire wealth when they die?

JingleBells

Actually, I now realize the idea is stupid.

I wasn't talking about executing the idea or anything but

bolbteppa

The internet was created by governments funded by taxes, all the people making money off it could never have made any of it or gotten anywhere near inventing it without decades of public funding, this is an old story, the idea rich people come out of a vacuum and do nothing but create benefits for everyone we should be thankful for is Gilded Age level thinking

Charlie

12:17 PM

It's not even stupid, and in a more idealised world something (maybe a little less drastic than that) would be beneficial. But there is absolutely no way you could pass a law mandating it, there is far too much influence from people who would stand to lose from it

ACuriousMind

@JingleBells The conflation of wealth and value is central to the narrative that capitalism is meritocratic, but not everything that is paid well has great value to society and not everything that has great value is paid well, unless you want to argue that health care and other infrastructural workers create less benefit to society through their work than an investment banker multiplying the wealth of already rich people.

bolbteppa

If you're not for a wealth tax you're for the creation of generational monarch's, basically again back to serfdom within a generation or so

JingleBells

@ACuriousMind Could you please clarify? The value of something is set by how much money people are willing to pay for it.

ACuriousMind

Ah, sure, if you believe there is no such thing as value other than monetary then you can't understand what I'm trying to say

bolbteppa

In the past 50 years, the workforce has doubled on top of population growth with the simple fact of women entering the workforce en masse, yet overall people make less than they used to, even though profits and productivity have reached stratospheric levels. But it's the owners who created all the value haha

Charlie

12:22 PM

I mean teachers get paid peanuts and they hold one of the most important jobs in all of society

bolbteppa

@JingleBells this is the kind of stuff a college-level course on politics would challenge your thinking on

Charlie

Or even just going to college honestly, for a lot of people getting out of their local bubble is an eye opener

ACuriousMind

@JingleBells I'm saying that e.g. the work of a health care worker that saves peoples' lives and eases their pains has moral value independent of its wages, and that most capitalist systems do not end up paying such a worker commensurate with that value.

JingleBells

@ACuriousMind I believe value is subjective and every living thing has different ideas of what's valuable and what's not. I may value cocaine more than health care and who are you to tell me I'm wrong. Capitalism allows every person, through every dollar spent, to cast their vote on what they value and what not.

ACuriousMind

...and those who are richer get to cast more votes?

Charlie

12:27 PM

How can I cast my vote on healthcare being valuable if I can't afford it lol

Also what you've said doesn't apply to public services that depend on government funding which repeatedly have their funding kneecapped

JingleBells

hmm, I guess I'm not a very good debater nor do I completely understand what I'm talking about. Everything is so complicated and my mind can't consider it all.

bolbteppa

Again the value of college is a place where you can learn how to defend and develop your ideas, challenge them etc

JingleBells

Please don't get me started on college.

Charlie

^ Yeah to this point it's not a coincidence that groups who believe the current system is fair also consider colleges to be "brainwashing" people

ACuriousMind

@JingleBells Yes, of course values are "subjective" and we need a way for societies to converge on which values they hold. Oddly enough, in the political sphere we have (tried to) invent a system where money isn't the means by which you vote, it's just...your vote, no matter how rich or poor you are. It's called democracy.

JingleBells

12:32 PM

@ACuriousMind honestly, I'd love to see you debate Ben Shapiro

bolbteppa

Bashing college, bragging about Shapiro, oh no :p

Charlie

Yeah jinglebells that is a dark path you're going down there lol

JingleBells

We'll see.

bolbteppa

Shapiro went to Harvard iirc

Charlie

ben "buzzphrase" shapiro is the bar against which all below the belt debating tactics are measured

yeah pretty sure he's a harvard trained lawyer right?

JingleBells

12:34 PM

@bolbteppa good for him

bolbteppa

The guy fueling all this anti-modern-society stuff went to Harvard, tried to write Hollywood comedies (lol) now bashing coastal elites etc

JingleBells

I have no intention of becoming a debater. I just seek to understand the world better (and no, the college is not the place for me to do that) I have not been to college but I've gathered enough evidence, opinion and feedback to make my decision.

Charlie

I don't see how higher education is not the place to "understand the world better"

ACuriousMind

@JingleBells The arguments I'm making here aren't original in the slightest. These are points critics of capitalism have made for ages.

JingleBells

Regarding college - I'm not paying $20,000 (or more) just so I can sit in a room and learn things I don't want to know or if I did, I could teach myself. And in the end, I get out with a piece of paper that can get me a job I do not want (in the long term). I have sufficient programming skills and I learn fast enough to manage to get a job while I make connections, debate topics with people, and learn different things from people who have achieved (or are trying to achieve) what I want

to achieve.

Charlie

12:41 PM

1. If the cost of college if off-putting to you, the very people whose ideas you seem to support hate the idea of removing college tuition. 2. You could of course teach yourself everything in principle, but an employer won't trust your word that you know what you're talking about over an official qualification from a university.

And fwiw leaving home, looking after yourself (usually for the first time) and being around people from all walks of life is just generally a valuable experience

JingleBells

I'm not going to argue about college. I have made my decision. If I fail, at least I would have gone with my heart and my mind, and not have trusted someone else to tell me what I should do. Our brains rely on information, and it's really interesting to see different opinions and sides being formed from each brain with only partial information considered.

Charlie

well it's been fun but now I'm hungry, cya guys

JingleBells

cya :P

BioPhysicist

@bolbteppa Lol "anti-modern-society".

JingleBells

humans are weird...

BioPhysicist

12:54 PM

More like "anti-anything that doesn't agree with me is evil- society"

JingleBells

@BioPhysicist are you socialist?

BioPhysicist

@JingleBells No, what made you think that?

JingleBells

donno, but everyone here seems to argue on the socialist side and i was wondering if there's at least one capitalist

BioPhysicist

oh. Well I suppose I support capitalism. Although the amount of time I have spent really considering economics is infinitesimal.

JingleBells

How can you ever know what is right or wrong when the essence of morality is questionable. Is it all about life preservation, happiness, reproduction, or something more, something else? Such a complicated system of ideas, opinions, sides... so much data that I don't know if I should even bother taking sides.

Some people may value technological progress more, other morality or fairness... humans. are. weird.

BioPhysicist

1:01 PM

@JingleBells True. If we don't start with a common set of values and morals then it will be hard to come to agreement.

ACuriousMind

@JingleBells That question is the essence of moral philosophy, and we've argued about it for hundreds of years.

Thousands, really, probably since language made it possible to express arguments about it

JingleBells

AI.

ACuriousMind

Your beloved AI is tightly coupled with moral issues, since the question is - if we really can create superintelligences, what should we teach them to value? How do we guard against terrible optima like the paperclip optimizer?

JingleBells

@ACuriousMind Then why do we even bother. I think our little brains are too f*cking stupid to even know what's going on. So much data and every person gets a little piece of it to form opinions on.

bolbteppa

Is there a way to do something like write the Maxwell action in terms of twistors or something to make them seem obvious

ACuriousMind

1:04 PM

@JingleBells Morality is not a question about what is that we could comprehend just by gathering enough data, but about what should be.

@bolbteppa I've tried plenty of times but I still don't know what a twistor really is supposed to be nor whether it's actually useful :P

bolbteppa

Yeah I realized yesterday I probably need to see them used in an action to make sense of them

BioPhysicist

I remember twistor from my childhood. Fun game

JingleBells

@ACuriousMind Sorry, please clarify. I don't think there are universal "moral" rules. I think morality is a set of "fair and unfair" innate guidelines that have helped preserve life for reproduction. It's evolution's way of saying "No one kills, everyone eats and creates babies".

ACuriousMind

@JingleBells "No one kills" is certainly not a maxim well-reflected in evolutionary history

Evolution is blind, it doesn't have an agenda. You may think that our desire for things like "justice" is something we acquired because it is evolutionary beneficial, but that doesn't explain which things are just or unjust.

JingleBells

@ACuriousMind I'm confused

ACuriousMind

1:13 PM

@JingleBells When people ask whether something is moral or not, they are asking a question different from whether it is evolutionary beneficial for us to behave that way or not. They are asking - what is the measure by which we should judge whether or not to take a certain action.

If you claim that that measure is whether or not it is beneficial to the survival of our species, then that is a moral stance in itself, and not a particularly helpful one because most actions in the context of modern society do not underlie any evolutionary pressure.

BioPhysicist

Plus you can get to some questionable morals if you think you are benefiting the evolution of your own species...

JingleBells

Sorry, my brain is really tired. I read but I can't contemplate. I'll come back later.

BioPhysicist

Ok, rest well :)

On an unrelated note, I just read the book "Recursion" for a second time and I believe it is my favorite book at the moment.

I guess I need to read it again based on the title lol

JingleBells

@BioPhysicist Hey! I've read that too!

I've even discussed it here before, it's really awesome

Physics Meta

1:33 PM

0

Q: Is "here a problem and my solution with my reasoning for each step, where has my reasoning gone wrong?" question considered off-topic?

I specifically used the formulation in the title from here, the only place I found to deal with this. But I do not know, what to take from it. To elaborate: This question arose from comments on this question. I know the "check my solution" questions are considered off topic. However, I think this...

bolbteppa

Twistors and the Pure Spinor Formalism - Lecture 1

►

Has a good intro to the GS superparticle at the timestamp

Charlie

2:30 PM

Is there terminology that distinguishes Lie groups that can have their entire structure defined by their Lie algebra and those that can't?

JingleBells

I want to debate in some topic but i dont know which

ACuriousMind

@Charlie What do you mean by "structure"?

Charlie

I mean using the exponential map on elements of the algebra to calculate the element at a particular point some distance from the identity

I'm wondering if there is a term to describe Lie groups for which any element can be written as an exponential map of some element of the algebra

At least as I understand it the exponential map cannot always do this, but it can at least do this locally in some neighbourhood of the identity

I am assuming the algebra is the tangent space at the identity not just some arbitrary point, at least that's what's been done in what i've seen so far

ACuriousMind

@Charlie Ah, there is no name for that other than "the exponential map is surjective"

Charlie

Ah ok fair enough, ty

ACuriousMind

2:39 PM

It is surjective for all compact connected Lie groups, and pretty obviously not surjective for disconnected groups. In the non-compact case, all bets are off :P

Charlie

Do we frequently run into those non-nicely behaved Lie groups in physics? I.e. the ones for which exp is not surjective?

JingleBells

I think it's really hard to argue in a chat form because you can't form a continuous discussion and it's difficult to express your thoughts with a keyboard. I feel like through chat the debate flow breaks

Charlie

@JingleBells Most colleges have a debate society maybe you should try that :P

ACuriousMind

@Charlie Sure, e.g. the Lorentz group (or its close relative $\mathrm{SL}(2,\mathbb{C})$). But what do you need surjectivity for?

Charlie

Nothing particularly, was mostly just curious

JingleBells

2:44 PM

@Charlie bruh.

Charlie

I don't know enough about Lie groups yet to distinguish between what is of interest in physics and what are perversely complicated mathematical cases lol

JingleBells

I'm not going to college.

Charlie

I also didn't realise the lorentz group was a Lie group so I guess I also just learned that :P

JingleBells

I'm tired of f*cking repeating that.

Charlie

@JingleBells I was pulling your leg

ACuriousMind

2:48 PM

@Charlie Almost any group you'll find in physics is either discrete (e.g. symmetries of a crystal) or a Lie group (e.g. rotations, translations, transformations that preserve some metric like Lorentz transformations,...)

Charlie

ok that's good to know

JingleBells

@Charlie I know but at this point, it's not fun anymore. It's really annoying when your father, your father's friend, your mother, people online, and everyone expecting me to go to f*cking college when I don't want to. Fortunately, I can make my own decisions, mistakes and successes.

Ahh, I'm sorry to all. I know I'm behaving weirdly today. I have a sore throat and my head hurts. I'm not feeling well, so please excuse my weird behavior... I don't know what I'm doing at this point.

Nevertheless, I'm still not going to college :P

Guru Vishnu

Hi everyone! Is the notation O16O (as highlighted below) for oxygen-16 a typo or some kind of standard?

Charlie

looks like a typo

unless it's meant to mean $O_2$ with one oxygen being $^{16}O$

idk if that's standard notation

I'm by no means and expert though maybe ask on chemistry.stackexchange.com if the answer is very important

Guru Vishnu

3:03 PM

Ok. Thanks for the clarification! I found it over here and this came in twice. I just wanted to ensure before making an edit.

Charlie

based on context I'd say it's a typo

Guru Vishnu

Ok. Done! It was a nice story, by the way.

It's during these times, I wish I were able to suggest an edit instead of making it directly.

Charlie

It would be nice if that were perhaps a feature on older posts, I can see why it would feel a bit weird making a correction to a post that is several years old

user434058

4:17 PM

Yo.

Charlie

hello

JingleBells

hello

JingleBells

4:47 PM

@ACuriousMind Yes, actually they do.

Charlie

Then why not weight votes in an election by someone's wealth?

actually nvm I don't want to get back into this

JMac

Reminds me of the idea that only land-owning men were allowed to vote. I'm sure a system like that would never lead to discrimination against women or the less well off...

JingleBells

@Charlie Yea, me too. It's much better in real life. (and no, I'm not going to college :D)

We're just small little stupid brains thinking we know how things are and how things should be.

I took a "Are you capitalist or a socialist" test and it said I'm in the middle, which is 8% of the total quiz takers. I now realize that with my limited information available and my limited brain capacity it doesn't make sense to make a decision. I refuse to put myself in any of those two boxes. I realized that people are different and the way people view the world and how it should be is different. Who am I to tell anyone how the world should work?

I'm just a small little brain with limited information to consider.

ACuriousMind

@JingleBells Everyone is

JMac

@JingleBells We're also the best we've got. We don't really have many alternatives besides using our "stupid brains" to know things and how they should be. I think it's a lot better that we try to understand and figure out how things should be using our "stupid brains" than just not trying.

Charlie

5:02 PM

I would take the "capitalist/socialist" online test about as seriously as an online IQ test

ACuriousMind

^also, that

JingleBells

@ACuriousMind sure

@JMac Yes, that's why people vote and we figure out what the collective wants.

@Charlie Soo, test is bs? I agree

JMac

@Charlie You're just jealous of my 20000 online IQ.

Charlie

The underlying point I was making is that you generally shouldn't label yourself and let the label tell you how to act

JingleBells

@Charlie The test reflects how I feel

That's why I'm constantly bringing this topic

Charlie

5:05 PM

@JMac Must be nice being able to move objects with your mind with an iq like that

JingleBells

I feel like I'm supposed to decide, but I now understand I don't have to take a side. I prefer capitalism, but I keep an open mind. I will take the world as people have chosen it to be, I adapt.

ACuriousMind

Not blindly accepting labels others try to force on you is good, but refusing to form an opinion because you think you're somehow not qualified to have one just doesn't work: You can think that because no one knows, we should abstain from any prescriptions to others at all - congratulations, you've discovered the libertarian strain of anarchism! - but that's still a political position like any other.

JMac

@Charlie It's a struggle. I'm obviously so smart, but any time I explain things to people using my super big brain, they act like I'm saying total nonsense. I don't understand!

Charlie

The struggle of the misunderstood genius is a tough one

JingleBells

@Charlie ya i can relate xDDDD

JMac

5:08 PM

@Charlie Worth noting that I had to design my own online IQ test to calculate my IQ because all the other ones I took were broken by my massive intellect and gave completely wrong numbers.

Charlie

Sometimes the best way to get something done is to just do it yourself

JingleBells

@ACuriousMind So, it's capitalism vs socialism vs not-qualified? xD

Charlie

that quote definitely applies to IQ tests

JMac

And space programs.

I expect to be on mars within a few weeks.

Charlie

:P

ACuriousMind

5:10 PM

@JingleBells No, it's "broad labels are used for reasons of realpolitik and beliefs are rarely a matter of binary choice in practice"

JingleBells

@JMac call elon

ACuriousMind

@JMac ...why was the test online if it was just for you? :P

JMac

@ACuriousMind You wouldn't understand...

JingleBells

@ACuriousMind Honestly, I'm losing your train of thought and all of a sudden I don't know what we're debating.

Charlie

It was so comprehensive and thorough it needed to be streamed from a server farm in China

This reminds me of the answers given to "what's it like to have a high IQ" on Quora. The responses are basically everything said so far, but unironically.

JMac

5:15 PM

@Charlie I was going to make a joke about me being unironic; but at some point Poe's law is going to kick in and I don't want to seem that unhinged lol.

Charlie

Have to draw the line somewhere :P

JingleBells

Since our brains are too limited to decide on their own, people vote and currently capitalism rules. People have tried socialism in the past, didn't work. Doesn't that tell us something?

Charlie

A lot of socialist countries in the past are actually socialism+ruthless autocracy

JingleBells

ok?

Okay, let's end this here.

Charlie

If Joseph Stalin emerged from the void and seized control of the United States and dragged it back to the middle ages it doesn't necessarily imply capitalism failed

JingleBells

5:27 PM

I thank you all for taking the time to discuss/debate this. I appreciate it. I've learned something, I hope you've learned something as well.

JMac

@JingleBells Does it really tell us anything? If our brains are too limited to decide on our own, why would the popular choice tell us anything about what is better? Maybe it just points to a common flaw in our thinking.

Charlie

@JingleBells If you want to just endlessly debate politics there are a million forums online that will entertain you for hours lol

JingleBells

@JMac Oh no, of course, it doesn't tell us anything. Who are they to tell you how the world should work.

@Charlie :O cool, i tried googling but debating through means of chat is really difficult and I'm honestly tired of it

Charlie

I'm sure you can find discord servers dedicated to the kind of discourse you want

JingleBells

People seem really sensitive about politics and I hope opposing parties don't hate each other, just like I don't hate you :)

Charlie

5:32 PM

here you go

JingleBells

Hey, let's debate abortion! xD

jk

Charlie

Some people fairly enough just roll their eyes at discussions about politics. To a lot of people it's just an endless conversation sink that ends in everyone begin unhappy

JingleBells

Let's debate feminism

@Charlie Yes, I figured that out :P

I don't think we did anything constructive here, but it opened my eyes to some things (which I don't want to state cuz they'll start another endless debate).

Charlie

Try those debate discords if you want to discuss polarising topics

JingleBells

Lol I'm entering a socialist discord server

this is about to get intersting

JMac

5:34 PM

@JingleBells To be fair, opening your eyes to more ideas could be considered very constructive when it comes to politics.

JingleBells

@JMac yes, I contradicted myself a bit there

Balarka Sen

"People have tried socialism" is a funny looking comment lol

I don't know why people say that

JMac

FYI odds are the people in those servers are very well versed in the ideas they are talking about, so you might be a bit overwhelmed by terminology and just get dogpiled by people who have already discussed the topics ad-nauseam.

Balarka Sen

Anyway what's cool in physics?

Charlie

BECs

JingleBells

5:36 PM

I think it's obvious that these debates are endless. People are different and feel different things. Just give them a vote and it's the best way to figure out what works and what doesn't.

Charlie

@BalarkaSen I recognise you from the maths chat what brings you here?

bolbteppa

Dirac's constrained quantization

Balarka Sen

@Charlie Boredom, primarily. I saw you asking some differential geometry a few days ago, that's why I thought of hanging out, actually.

Might see more questions I could take a dig at

Charlie

fair enough

Balarka Sen

@JingleBells u could debate about democracy too. we cant hold a vote to see if people want democracy though, because that'd be biased!!

bolbteppa

5:38 PM

10

Q: What is kappa symmetry?

On page 180 David McMohan explains that to obtain a (spacetime) supersymmetric action for a GS superstring one has to add to the bosonic part $$ S_B = -\frac{1}{2\pi}\int d^2 \sigma \sqrt{h}h^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X_{\mu} $$ the fermionic part $$ S_1 = -\frac{1}...

quantum-field-theory string-theory supersymmetry symmetry superspace-formalism

is also very cool

JingleBells

my england is very olequentaly spoken

Balarka Sen

@bolbteppa uh oh Mr Urs is here

bolbteppa

Yeah I would ignore that stuff

Balarka Sen

lolol

super $L_\infty$-algebras is where its at

JingleBells

I entered a rightist discord server and some guy is telling fascism is best..

at least we can all agree hitler can enjoy his place in hell, right? ...... right?

Balarka Sen

5:45 PM

yikes @ACuriousMind

Charlie

What's the deal with Urs?

Balarka Sen

i just came to see what the physicists were up to

instead i see socialism and mr adolf

@Charlie he's a weird guy

Charlie

oh

JingleBells

@BalarkaSen my apologies, i've moved to a discord server

oh, hitler != fascism, hitler == national socialism (nazi). Bruh I'm learning stuff

Balarka Sen

Lmao

Charlie

5:49 PM

as you can see we stay strictly on topic here

JingleBells

but ofcourse

JMac

@BalarkaSen To be fair, this is a general chat for the Physics SE; but it doesn't mean the chat has to be about physics. It's likely you'll find people familiar with physics in the chat, but the chat itself can be about anything (within reason and within the realm of respecting others)

Balarka Sen

thats cool with me

satan 29

hi all

JingleBells

omg discord is so f*cked up

JingleBells

6:08 PM

I now appriciate this chatroom after having gone to a dicord politics room

everyone is saying the n word and are posting pictures of dogs and pot...

Balarka Sen

Lmao

Charlie

Do the structure constants of a lie algebra corresponding to a lie group uniquely identify it?

My book didn't realise go into that much detail on them and what I can find online is quite advanced

ACuriousMind

@Charlie no, different lie groups can have the same algebra, but they're all quotients of the same universal cover

@BalarkaSen I've only got half an eye on this room right now :P

Balarka Sen

That's the only interesting way two Lie groups can have the same Lie algebra, basically.

Charlie

Ok I'm happy that different groups can have the same algebra since they can locally be the same

Balarka Sen

6:18 PM

That's exactly the right idea. An explicit example is something you guys adore, SU(2) -> SO(3).

Charlie

oh wait so the structure constants don't uniquely label the group but they do uniquely label the algebra

This actually leads me onto another point of confusion I've been trying to work through. Why any particular choice of basis in the algebra is preferred over any other, is it just a matter of convenience?

Balarka Sen

Structure constants only determine the universal enveloping algebra, not the algebra itself I believe

Charlie

At least as far as I've seen in physics we choose for instance the Pauli matrices to generate (I can't immediately remember which) a lie group

unless there's just a canonical choice

ACuriousMind

@Charlie there are special choices of bases that are useful for certain things, but if you haven't come across them there's little point in telling you about them

Charlie

I'm less concerned about specific examples just yet anyway, was more just curious why it is sometimes stated that "x,y,z are the generators of this lie group" rather than that they are just a convenient choice

but I guess that makes sense then

I need to go and look at texts on specific groups now anyway I'm done with the section of this book

Balarka Sen

6:32 PM

I actually have no example of nonisomorphic Lie algebras $\mathfrak{g}, \mathfrak{h}$ such that $U(\mathfrak{g}) \cong U(\mathfrak{h})$ as associative algebras.

I googled. This is an open problem (in characteristic zero)!

11

Q: On the isomorphism problem of enveloping algebras

Let $\mathfrak{g}$ and $\mathfrak{g}'$ be Lie algebras. It is known that if $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as associative algebras, then it is not necessarily true that $\mathfrak{g}\cong \mathfrak{g}'$ as Lie algebras. I am looking for examples such that $U(\mathfrak{g})\cong U(\mathfr...

rt.representation-theory lie-algebras hopf-algebras monoidal-categories

bolbteppa

6:48 PM

Physics motivates the advancement of math again... :p

Mike Miller

@Charlie They certainly uniquely determine the Lie algebra. A linear map $A: V \to W$ is determined by its matrix representation, and the structure constants are just saying what the map $[-, -] : \mathfrak g \otimes \mathfrak g \to \mathfrak g$ is, as a matrix.

Equivalently, given any bilinear map $f: V \times V \to V$, you only need to know what $\langle f(e_i, e_j), e_k\rangle$ is to reconstruct $f$. You can prove this by hand; it comes down to the definition of a basis.

Balarka Sen

(The point is the second you remember more structure on U(g), like the graded presentation, or even the bialgebra structure, you get back g -- I ended up asking a good question accidentally that's not really relevant. Mike pointed this out to me)

Charlie

7:03 PM

Ok thank you

some of this is too advanced for me yet, but hearing that the algebra is uniquely determined by its structure constants is helpful

bolbteppa

The structure constants change as a (2,1) tensor under a change of basis, writing out $[X_i,X_j] = f_{ij}^k X_k$ and $X_i = U_i^j Y_j$ it's easy to see, you'd expect it to be basis independent but this tensor still defines the Lie algebra

julien surel

structure constants determine the lie algebra completely, that's why they are called 'structure' constants.

Charlie

names can be deceiving, especially in physics :P

bolbteppa

They determine the structure of the Lie algebra (sorry, read that as 'then why are they called...')

Balarka Sen

I mean the constants (which are numbers) cannot possibly determine the algebra, you need to know the basis - knowing both recovers the Lie algebra yeah

Or making the constants tensorial like bolbteppa suggests

julien surel

7:14 PM

that's true, kind of. although there is a natural way to build a basis based on the Cartan subalgebra as far as I know

bolbteppa

For a semi-simple Lie algebra only right

Balarka Sen

what I was asking is essentially the following; write $[X_i, X_j] = f_{ij}^k X_k$ and look at the associative algebra generated by $x_i$'s, modulo relations $x_i x_j - x_j x_i - \sum_k f_{ij}^k x_k = 0$. This is some algebra $A$, depending on the Lie group, and are isomorphic as algebras if you choose different bases.

can the Lie algebra be recovered from $A$ without knowing nothing more about the structure of $A$, except the associative algebra structure? this is unnecessarily complicated and irrelevant

but thats somehow the open problem. shrug

@juliensurel tell me more, i dont think i know this

ACuriousMind

The "Cartan-Weyl basis" of the mathematicians is what physicists will recognize as "ladder operators"

You have the generators of the Cartan subalgebra and the rest of the generators acts as raising/lowering their eigenvalues

julien surel

I don't remember the details, but Cartan subalgebra corresponds to the maximal torus of the corresponding Lie group, it's the maximal abelian subalgebra of the Lie algebra.

Balarka Sen

Ah ok I didn't know that's used interchangeably with maximal torus

ACuriousMind

7:21 PM

well, it's not *interchangable" - it's the manifestation of the maximal torus in the algebra

I don't think anyone calls the torus the "Cartan torus" when talking about the torus in the group

Balarka Sen

yeah I realized after I wrote the comment

didnt know that was the name for this thing, i used to just call it the lie subalgebra of the torus

@ACuriousMind How do you think of the root vectors like that?

ACuriousMind

@BalarkaSen The Cartan subalgebra is Abelian (at least in the semi-simple case over the complex numbers physicists are usually interested in), so in any representation, it is represented by commuting operators, so you can give every representation space a basis of common eigenvectors of this algebra

Balarka Sen

Ah

I'd never thought about that

ACuriousMind

also, the subalgebra is diagonalizable w.r.t. the adjoint action of the algebra on itself as a special case of this, so you can also decompose the algebra itself as eigenspaces of the Cartan operators - the eigenspace with eigenvalue 0 is the Cartan algebra itself, and the eigenspace with the eigenvalues $(\lambda_1,\dots ,\lambda_n)$ acts as shifting the eigenvalue of the $i$-th Cartan operator by $\lambda_i$ in every representation

Balarka Sen

that makes sense

julien surel

7:33 PM

I also learned all of these recently, I needed that to learn about infinite dimensional lie alegbras Like Virasoro Algebra of CFT.

Balarka Sen

@ACuriousMind So you let $T$ act on $\mathfrak{g}$ by the adjoint action, simultaneously diagonalize, take a basis of eigenvectors - the null guys are in $\mathfrak{t}$, the other guys are the root vectors, yeah?

i never understood this root business but that is surprisingly clean

ACuriousMind

the roots are basically the $(\lambda_1,\dots,\lambda_n)$

Balarka Sen

ahh

ACuriousMind

You formally define them as living in the dual of the Cartan algebra to be independent of scaling issues, but that's the essence

Balarka Sen

Ok understood, thanks

for some time i was trying to see if i could guess what roots are from the biquotient $T \backslash G /T$

JingleBells

7:57 PM

Hmm, I don't see the problem with the rich getting to cast more votes on what society values. After all, most rich people will not all of a sudden go and spend their wealth on dog poop. And even if they do, the result (whatever it is) will get wiped away quickly as it's all short term.

And also, for them to be rich, they have created value to society and therefore know what society values and are unlikely to go and spend they money on some random thing...

So ACM I'm not sure what point you were making.

It's in fact unfair to treat some lazy ass stupid moafucka the same as someone else who is smart and wants to create and innovate.

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