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12:00 PM
Also, where can I read about Casimirs in math literature? I remember looking for it in the rep. thy. books I have, but none of them mentions them...
 
@JohnRennie didn't turn out amazing, but here it is imgur.com/gallery/FLEgX
 
@Danu They are elements of the center of the universal enveloping algebra, maybe your books just don't call them Casimirs
That these are useful for classifying/detecting irreducible representations is also nothing deep, it's just Schur's lemma
 
I'm not sure I've even seen universal enveloping algebras much.
 
@0celo7 hmm, that link gives me a redirect error
 
Also, I think the Wiki article already gives the relevant things one would need to know
You'll still need to figure it out for $\mathrm{Sp}(n)$ specifically, though, but as I said I can't help with that
 
12:07 PM
@JohnRennie Here's the picture in the link
@0celo7 Imgur links can be directly pasted here
 
@2017 I'm on mobile
I think people assume I'm retarded when I'm on mobile
 
@Danu What you may in general observe is that given two groups and representations $(G,\rho),(H,\sigma)$, the representation of their product is naturally $\rho\otimes\sigma$, so this explain the occurence of the tensor product in what you wrote.
 
Ooh that looks good :-)
 
@0celo7 So? Can't you copy paste from mobile?
 
@2017 too much work
 
12:09 PM
Something en croute, though I can't tell what from the picture.
 
@0celo7 I find the "Cat?" caption a bit disturbing :)
 
I have to paste the Imgur link into chrome, load the image, click on the image, get the link, then post it
 
@ACuriousMind I was confused since I thought it had to be the direct sum
 
@JohnRennie beef Wellington
 
@0celo7 Aha!
 
12:09 PM
One with mushrooms, one without
 
Excellent :-)
 
Actually it's the diabetes bomb. It's pretty unhealthy
Well...I guess filet is lean
 
Is it? There are carbs in the pastry but there isn't that much pastry.
 
There's a lot of eggs. But it's also a lot of food, so meh
Then there's olive oil, but not much
 
@Danu Well, that's also possible, but not really natural, since it is not irreducible even if the original ones are irreducible.
 
12:12 PM
So maybe it's a diet food!
 
Provided you eat some green veg with the BW I'd have thought it was a pretty balanced meal.
 
@ACuriousMind I don't believe that it should naturally be the tensor product, since that might inadvertently might make the natural thing built out of two injective representations not be injective, as indeed happens in my example.
 
@JohnRennie roasted potatoes
 
Since the tensor product of the fundamentals of $Sp(n)$ and $Sp(1)$ in fact factors through $Sp(n)Sp(1)$
 
It looks spot on though. The pastry has cooked beautifully.
 
12:13 PM
A salad would have been nice, but it didn't happen
 
@ACuriousMind Yeah, exactly that was my issue with the direct sum.
 
@Danu Faithfulness is a much less natural criterion than irreducibily :P
 
Is it? :P
 
Recall that there are finite-dimensional groups which admit no finite-dimensional faithful representations at all!
 
Injectivity is probably in my top 3 of most important properties of anything :P
"recall" :D
I'm recalling from a past life
 
12:14 PM
@Danu Surely you've heard "not every Lie group is a matrix group", no?
 
Yeah, I know
But I wouldn't be able to tell you
any examples
 
And faithfulness doesn't tell you anything interesting about the representation - in particular, it doesn't help with the main task of representation theory, which is to find irreps and then decompose everything into them
 
@JohnRennie it's a little overcooked, and I don't know how to make it firm
the pastry kind of falls apart
 
@Danu They're weird, generally. Cover of the symplectic group, non-compact complex form of $E_8$, that sort of thing
 
12:16 PM
Morning
Say, why doesn't this site have a vector logo, like all the other SE sites?
 
alright
so the universe doesn;t have a center
because
 
That looks fine. The pastry always falls off when you cut into a beef wellington. Unless you make the pastry very thick, in which case the texture is rubbish.
 
the surface of a ball doesn't have a centre
 
@MartianCactus Well that's assuming the universe is closed...
 
@JohnRennie The beef is just too juicy, it makes the bottom of the pastry soggy
 
12:17 PM
but a sphere does have a centre
not on the surface
but in the core, right?
and what do you mean by "the universe isn't closed" ?
 
@MartianCactus a sphere does not contain the center of the ball it bounds
 
@0celo7 ah, OK, yes that always happens
 
@MartianCactus OK, lemme explain. There are multiple shapes that a universe can have, depending on the variables we use and how we model it
Our real-life universe likely isn't shaped like the surface of a sphere; rather, it's probably flat like a sheet of paper...except 3D
 
so like a cuboid?
 
Yep :)
Here's the catch: it probably extends infinitely
 
12:20 PM
isn't universe a sphere in 3 dimensions tho?
 
@SirCumference trying not to troll
 
well if it extends infinitely
then it has no shape?
no wait
its a cube
 
@MartianCactus Yep, and the cube extends infinitely
 
Now lemme ask a question: can there be a center in an infinitely big cube?
 
12:21 PM
then how can it have
a shape?
if its infinite?
but how can it even have a shape?
and if its infinite then how is it expanding?
 
@MartianCactus When we say "shape" of the universe, we really mean this: if we draw a triangle between 3 coordinates in space, will it have exactly 180°, more than 180°, or less than 180°?
 
what?
 
Infinitely-extended things can have shape, but not in the same way finite objects have. For instance, there is no appreciable difference between an "infinitely-extended cube" and an "infinitely-extended ball", they're just both $\mathbb{R}^3$.
 
@MartianCactus I'll get to that
 
a triangle has exactly 180
 
12:23 PM
@MartianCactus The triangles you've seen in high school geometry, yes
But if you draw a triangle on the surface of a ball, it will have more than 180°
If you draw it on the surface of a potato chip or saddle, it'll have less than 180°
 
r cube/
woah
really?
 
Yep :)
You've seen Euclidean geometry, which talks about the geometry of flat surfaces
 
oh
yeah
 
Non-Euclidean geometry describes the geometry on curved surfaces, and different things happen. For example, parallel lines can meet on a sphere.
 
@MartianCactus Take a sphere and draw a triangle between the north pole, and two points on the equator that are a quarter of the circumference apart. All angles of this triangle are right angles.
 
12:25 PM
how?
parallel lines cant meet on a shpere either tho
 
@MartianCactus You draw all the lines from pole to pole. They'll be parallel at the equator.
So anyway, the shape of the universe basically refers to the curvature of space.
I gtg, I'll be back in 20 mins
 
@SirCumference Now that I've never found helpful because it depends on knowing the proper generalization of the notion of parallel. Few people would guess it's "part of a great circle" and not "lines that don't intersect".
 
Ugh, it would took me a while to fix this, because the reason why it is off topic is because both questions are WAY too broad, so broad that just splitting them up into two questions is still not enough to make it on topic according to Joseph Weismann
1
Q: How are beliefs restricted by an objective reality, and do unreal beliefs can only cause real impacts indirectly?

SecretEver since reading this NewScientist issue on how even if for some beliefs that despite being based on falsehoods, can have actual impacts to the world via cultural influences, I started to wonder about the nature of belief, and by extension, unreal entities or concepts that have real impacts. C...

 
what do you mean by "parallel lines aae part of a great circle?"
 
@MartianCactusOK I can be back for a few mins
So as I said, we've measured that our universe is flat — that it follows Euclidean geometry, since 3 spatial coordinates make a triangle with exactly 180°
A "closed" universe is one in which a triangle of spatial coordinates would have < 180°. That would be shaped like the surface of a sphere.
 
12:30 PM
@Secret I've voted to re-open
 
@MartianCactus Now, let's assume the universe is infinite. So my question for you is: can there be a center in an infinitely big cube?
 
@Kenshin thanks, but no you missed the point. I have actually talked with the closers and the above reason is the precise reason they gave me on how to fix it
 
stuff them tho
 
So without some heavy fixing it is not going to reopen so easily
 
the question reads fine to me
 
12:32 PM
I am currently splitting up the question, and see if there are leads to make it more focused
 
their opinion is wrong
 
no
well yes maybe
 
@MartianCactus Right, since every point can be called a center.
 
but no
any point we take
ya
 
That means we won't have a definitive center
 
12:32 PM
every point we take will be a center
yep
 
Yep :)
 
but if its infinite
then it cant be of any shape either?
because its infinite
 
@MartianCactus Well, remember: shape merely refers to the curvature of space.
We've measured the universe to be Euclidean, meaning that a triangle of 3 spatial coordinates will have exactly 180°
 
wait what?
i thought euclidian geometry was 2D
but universe is 3D
 
@MartianCactus Euclidean geometry can be extended into 3D. Remember, all we said is that Euclidean geometry doesn't work on warped/curved surfaces.
 
12:34 PM
@Kenshin You do not have enough reputation on Philosophy to do so, what are you talking about?
 
oh
so like no curve
 
= euclidian
oh
 
@ACuriousMind uhhh I do have enuf double check
 
Nothing distorting
 
12:35 PM
so we have a flat cube
which is infinite
 
@MartianCactus Yup
@MartianCactus So, now to explain expansion. This is the cool part.
 
wait wait
what do you mean by
 
@ACuriousMind you know the rep requirements in beta sites are lower don't you?
 
"shape refers to the curvature of space" ?
 
@MartianCactus Well, we say the shape of the universe is like a sphere if a triangle of 3 spatial coordinates is less than 180° degrees
We say that it's shaped like a saddle if it has more than 180°
 
12:36 PM
@Kenshin You have 756 reputation, and the reopen privilege is granted at 3000 rep since it's a graduated site.
 
oh so
 
@Kenshin Philosophy is not a beta site.
 
in advanced level stuff
 
@ACuriousMind yoiu know that you responded to me saying "I've voted to re-open"
 
we define shape according to the sum of the angels of a triangle made on their surfacE?
 
12:37 PM
Sorry, I meant "Well, we say the shape of the universe is like the surface of a sphere if a triangle of 3 spatial coordinates is less than 180° degrees"
 
and you know the requirements to vote to re-open is certainly less than 3000
 
@MartianCactus Yup
 
@Kenshin What? No, it isn't
You cannot "vote to reopen" with less than 3k rep
 
Wrong because I already did
 
less than?
 
12:38 PM
That re-open vote there is mine
 
i thought on a sphere its every angle = 90 degree
 
@MartianCactus Yeah, a triangle on the surface of a sphere will have less than 180°
@MartianCactus I'm sorry?
 
@ACuriousMind it says 500 reputation needed to vote to re-open
 
@Kenshin Yes, I just found that
Interesting
It is a graduated site with beta-level rep requirements
 
wait what?
curious mind told me that if yuo kay out a triangle on the surface of a sphere veeyr angle is 90 degree
 
12:40 PM
@ACuriousMind also odd the banner looks like it is still in beta
 
@SirCumference Question for you
 
and the sum 90 + 90 + 90 = 270
 
@0celo7 I'm in the middle of something
 
Suppose you're living in a universe with $K\ne 0$
 
which is more that 180
 
12:40 PM
@SirCumference it will be more than 180
 
@0celo7 Ok
 
How would you measure the angles of triangles?
 
@Kenshin It has no "beta" on it - the site just hasn't received its custom design yet for whatever reason
 
yep
 
I imagine on Earth scales the curvature is negligible
 
12:40 PM
@0celo7 I dunno how they do it
 
They? Has anyone done it?
 
I just know they figured it out with the Planck satellite
 
proof?
 
Planck was a space observatory operated by the European Space Agency (ESA) from 2009 to 2013, which mapped the anisotropies of the cosmic microwave background (CMB) at microwave and infra-red frequencies, with high sensitivity and small angular resolution. The mission substantially improved upon observations made by the NASA Wilkinson Microwave Anisotropy Probe (WMAP). Planck provided a major source of information relevant to several cosmological and astrophysical issues, such as testing theories of the early Universe and the origin of cosmic structure; as of 2013 it has provided the most accurate...
 
@Kenshin Aha!, the rep threshhold increases will come with the design
 
12:41 PM
@ACuriousMind did my bw not even make you hungry :(
 
so yeah
 
@ACuriousMind isee, better make the most of my priviledges quick
 
if you lay out a triangle on sphere the sum of its interior angles is more than 180
 
@SirCumference Wiki says it measured the CMB, which is what I recall
 
@0celo7 I've been hungry even before you posted it
 
12:42 PM
control F "triangle" gives nothing
 
i think its 270
 
I'm also still concerned about the "Cat?"
 
that what @ACuriousMind said
that every angle is 90
 
@ACuriousMind the cat is still alive, I can give proof
 
@0celo7 That's relevant, I think
 
12:42 PM
when you lay out a triangle on the surface of a sphere
 
WMAP did the same thing
The Wilkinson Microwave Anisotropy Probe (WMAP), originally known as the Microwave Anisotropy Probe (MAP), was a spacecraft operating from 2001 to 2010 which measured temperature differences across the sky in the cosmic microwave background (CMB) – the radiant heat remaining from the Big Bang. Headed by Professor Charles L. Bennett of Johns Hopkins University, the mission was developed in a joint partnership between the NASA Goddard Space Flight Center and Princeton University. The WMAP spacecraft was launched on June 30, 2001 from Florida. The WMAP mission succeeded the COBE space mission and...
 
@MartianCactus No, I gave a very specific triangle that has 270 angle sum. not every triangle on the sphere has
 
> The current expansion rate of the universe is (see Hubble constant) of 69.32±0.80 km·s−1·Mpc−1. The content of the universe presently consists of 4.628%±0.093% ordinary baryonic matter; 24.02%+0.88%
−0.87% cold dark matter (CDM) that neither emits nor absorbs light; and 71.35%+0.95%
−0.96% of dark energy in the form of a cosmological constant that accelerates the expansion of the universe.[7] Less than 1% of the current contents of the universe is in neutrinos, but WMAP's measurements have found, for the first time in 2008, that the data prefers the existence of a cosmic neutrino backgrou
 
@ACuriousMind but is sum of every angle more than 180 on a sphere-triangle-shape?
 
Yes.
 
12:44 PM
@SirCumference so lets recap
 
The smaller the triangle, the less it deviates from 180°
 
@MartianCactus OK
 
when we make a sphere-surface-triangle
sum of all angles is more than 180
right?
and
 
Oh god, have I been saying that triangles on a sphere have less than 180°?
I'm tired
 
12:45 PM
I meant more than...
 
WOw
 
more than* ??
 
Great moves Sir Cumf, keep it up
 
12:46 PM
I'm fducking stupid
And saddles and potato chips have less than
 
and universe is an infinite cuboid
 
@MartianCactus By our measurements, most likely
 
which is straight surface
 
@MartianCactus Pretty much
 
and euclidian can be used on it
it is infinite and still a cuboid
 
12:47 PM
Yep
 
because in high level stuff
shapes are defined by curvature
and what the sum of angles in the triangles made on their surfaces will be
 
@MartianCactus Well, with regards to the universe at least, yes
 
and this new definition for shapes started to be used because we needed to give infinity a shape?
 
We'd say a 2D figure has a shape, no matter the surface on which it's drawn
A pentagon has a shape, but no curvature
Only when we refer to the universe's shape do we mean curvature
 
oh so in 3D
we define shapes with curvature
 
12:49 PM
@MartianCactus I guess
 
wait, doesnt 2d have curvature too?
 
I'd say for now, just keep it with regards to the universe
 
ah
ok
so shape - curvature
FOR UNIVERSE
right?
 
ok now lets start with the cool expansion stuff!!
 
12:50 PM
Yep :D
OK, so when you hear "the universe is expanding", you probably imagine a ball growing bigger and bigger
It's not actually like that
 
I must be stupid, why can't the universe be RP^3
 
@0celo7 Oye, hold questions
 
$\Bbb RP^3$ isn't orientable
 
Who said the universe has to be orientable?
 
Einstein
 
12:53 PM
Fermion physics
"RPn is orientable iff n+1 is even, i.e., n is odd."
Oh
 
@MartianCactus Rather, when we say "the universe is expanding", we really mean "space is expanding". By this, we mean that the distances between almost all matter are increasing
 
I guess not?
 
Hmm.
 
I put almost so no one jumps on me before I get my point across
 
oh
but most llustrations show universe as a ball
 
12:54 PM
Here's an easy analogy: imagine you are walking your dog. Suddenly, the ground begins expanding between you. You and your dog will separated and continue receding away from each other.
 
and michio kaku said that universe is a ball
 
@MartianCactus Yeah, ignore those
They're often misleading
 
@MartianCactus Also misleading
 
you said that
 
12:54 PM
But read that analogy
 
The universe isn't a ball
The universe is everything
 
^ignore that
 
don't ignore it
it's true
 
12:55 PM
@Kenshin He's asking about the shape of the universe
 
the universe occupies literally everything and everywhere it is shapeless
 
@Kenshin Oye...
 
Geodesics may be curved, but the universe itself is not
 
I just got through explaining this
 
so if universe is the surface of a balloon and we put 2 dots into it
then we blow into it
 
12:56 PM
Just because a geodesic is curved doesn't mean the universe is curved
 
the space betweeen the dots will expand
 
I have to go, but I'll be back later
 
alright!!
 
@MartianCactus Here's an animation I made to help you get it
 
but this guy said that the universe is shapeless
 
12:56 PM
 
@MartianCactus I can answer your questions
The universe does expand
 
@Kenshin Wasn't I the one who trained you?
 
and it is analgous to a baloon
@SirCumference uhh no
 
@Kenshin No, no...
 
@SirCumference my university trained me
 
12:57 PM
Don't use that analogy, it's bad
 
analogies aren't 'bad"
 
Yes, that one is
 
the baloon analogy?
 
humans think in analogies many great scientists including Feynman used them
 
People are gonna ask "what happens if the balloon moves", or "how can we get inside the balloon"
 
12:57 PM
The balloon analogy is very fitting that is why Suskind uses it
 
The analogy falls apart
 
analogies aren't designed to never fall apart
 
they are designed to be analogies for particular points, not exact replications
 
I can't be here for long
 
12:58 PM
laterz
 
@Kenshin Yes, but you're better off ignoring that analogy
 
I'll take it from here
 
It brings up more questions than answers
 
noice animation tho :D
 
12:59 PM
I disagree
 
I generally use my dog analogy
 
The balloon analogy is just an analogy @MartianCactus
 
i will read this first
 
but it highlights an important point
 
12:59 PM
232
A: Did the Big Bang happen at a point?

John RennieThe simple answer is that no, the Big Bang did not happen at a point. Instead it happened everywhere in the universe at the same time. Consequences of this include: The universe doesn't have a centre: the Big Bang didn't happen at a point so there is no central point in the inverse that it is e...

 
@MartianCactus Lemme just say one thing
 
i will read this
 
yes that is a good read Martian
 
and then come back to chat
 
Look at the animation I made
 
12:59 PM
cos the universe is always infinite
 

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