someone trying to make you do representation theory? just say no, they cannot legally make you learn representation theory
@simeondermaats
Жыл бұрын
yess! I've been waiting for this one, lovely to see you upload again. Thanks for the high quality material and thanks for providing it for free!
@scottmiller2591
Жыл бұрын
Reeally hoping we get some exponential map stuff, although it sounds like no.
@JosBergervoet
Жыл бұрын
And in style!
@Happy_Abe
Жыл бұрын
@20:50 the first entry in the cross product should say a2b3-a3b2
@homerthompson416
Жыл бұрын
That Witt algebra video was really something. Very excited to see this series about math's greatest lie.
@Juniper-111
Жыл бұрын
aaaaa no way! I just started taking a class on Lie algebras and now you're doing a series on them!
@PunmasterSTP
7 ай бұрын
How'd your class go?
@numankaya6812
2 ай бұрын
yemin ederim ki sabah akşam sadece ve sadece bunlara çalışacağım
@kickuchiyo8586
Жыл бұрын
Great timing! I recently picked up a book called Lie Algebras in Particle Physics, so this series will be super helpful for me
@andreashuaman2041
11 ай бұрын
can you pass methat book ?
@enpeacemusic192
Жыл бұрын
Omg I’ve been wanting to learn more about Lie groups/algebras for a longgg while now, so this is amazingg
@ak47tetris84
Жыл бұрын
Years we have waited
@Bolvar_
Жыл бұрын
So happy too see new uploads on the channel, and a great topic too !
@jplikesmaths
Жыл бұрын
I have waited so long for this series! Thanks Dr Penn!
@wilderuhl3450
Жыл бұрын
I’m excited for this series
@ModuliOfRiemannSurfaces
Жыл бұрын
I’ve been needing to learn some of this for years so thank you for putting this up and forcing me 😌
@riccardoguidotti8770
Жыл бұрын
Really awesome content! I can't wait for the next video about Lie algebras, please upload it soon🙏
@lionskenedi4247
4 ай бұрын
sir keep up the good work. I have learnt a lot from you from both channels.
@mrl9418
Жыл бұрын
I think there's a plus instead of a minus in the very last formula you say. Lots of videos on Lie Algebras would be really interesting !
@malawigw
10 ай бұрын
I love lie groups and lie algebras and I love this series!
@andreapaolino5905
11 ай бұрын
It seems to me that the first proposition shouldn't be an iff condition, rather it should simply determine the second implication, aka, "If char(F) =/= 2 then [y,x] = -[x,y] implies [x,x] = 0 (if char(F) =/= 2, anti-commutativity implies alternating property)". This is because bilinearity + alternating property already imply anti-commutativity in ANY characteristic (0 = [x+y,x+y] = [x,x] + [x,y] + [y,x] + [y,y] = [x,y] + [y,x], so [y,x] = -[x,y]). Let me know in case I'm not following the argument
@lexinwonderland5741
Жыл бұрын
AAAAAAAAAAAAAAAAAAAAAAA IVE BEEN WAITING ON THIS SERIES FOR LITERAL YEARS DR PENN YOU HAVE NO IDEA HOW EXCITED I WAS TO SEE THE NOTIFICATION FOR LIE ALGEBRAS ON MATH MAJOR!!!!!
@zhuolovesmath7483
Жыл бұрын
So glad to see you again!!!
@Budha3773
10 ай бұрын
Having trouble with 4... I computed it to be Jx+(x^T)J
@briangronberg6507
Жыл бұрын
Thanks, Professor
@KhaledRadwan-ku2bh
Жыл бұрын
And finally Lie algebra, please give some time to the isomorphism between SO(3), SU(2) and 3 and 2 spheres. In addition, give some time to the concepts of connectedness, simple connectedness and compactness.
@mMaximus56789
Жыл бұрын
I would love to see calculus on Lie Groups/Lie Algebras
@rand_-mk5lb
Ай бұрын
Much easier than I thought. Random question, is gluing computational? Thanks.
@johnsalkeld1088
Жыл бұрын
Nice - i believe a derivation can have a different left and right action - i recall the Fox free differential - it (for the left action group ring) defined the right action as. The action after the trivialised of the group ring element to the ring only to D_x (vw) = v D_x(w) + D_x(v) tr(w).
@m9l0m6nmelkior7
Жыл бұрын
I love that you do this video !! But I have a question, at 3:05, why does [x+y ; x+y] = [x;x] + [x;y] + [y;x] + [y ; y] ?? is this because of bilinearity ?
@szymonkauzny2931
Жыл бұрын
Yes, [x+y, x+y]= [x+y, x] + [x+y, y] and you do the same with first coordinate and get exactly that.
@andreashuaman2041
11 ай бұрын
what a bout de char(F)/=2 c ondition?@@szymonkauzny2931
@NoahPrentice
Жыл бұрын
cool!
@sayanjitb
2 ай бұрын
in the definition itself why didn't you mention bi linear property of the lie bracket?
@PunmasterSTP
7 ай бұрын
If I said this wasn't an amazing lecture, I'd be Lie-ing!
@nunoalexandre6408
Жыл бұрын
Love it!!!!!!!!!!!!!!!
@bo77om
Жыл бұрын
yeeeee Lie algebras finally!
@devgumdrop3700
11 ай бұрын
For the part on derivations, with the lie bracket defined by [D_1, D_2] := D_1D_2 - D_2D_1, can anyone elaborate a bit more on the underlying algebraic structure here? Unlike other purely algebraic vector spaces I've seen before, there appears to be another relationship between the vectors and the field by the fact that the field elements can be inputs to the vectors. It appears to me like derivations are special types of functions. So is this space a function space, or something?
@Jason4195
Жыл бұрын
This is great! Do you have a recommendation of a book (preferably a Dover book) to go along with this? Thanks!!
@briansmith7458
11 ай бұрын
3:10
@LASLOEGRI
3 ай бұрын
2 min 20 sec into this and I’m already lost. It seems to be removing understanding from my mind.
@khaledchatah3425
Жыл бұрын
Question: What does the trace of matrix (matrix of a linear transformation) mean i get that it is the sum of the entries of the diagonal but what does it mean
@schweinmachtbree1013
5 ай бұрын
it is also the sum of the eigenvalues
@geraldpysniak6228
9 ай бұрын
if he works hard enough teaching may be possible
@hyperduality2838
Жыл бұрын
Commutators = two paths. Abelian (commutes, symmetric, Bosons) is dual to non abelian (non commutes, anti-symmetric, Fermions). Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung. Vectors are dual to co vectors (forms). Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Homology (syntropic) is dual to co homology (entropic) -- same is dual to different. Injective is dual to surjective synthesizes bijective or isomorphism. Subgroups are dual to subfields -- the Galois correspondence. "Always two there are" -- Yoda.
@FranFerioli
10 ай бұрын
There should be a way to measure how far down the rabbit hole you are when watching a given math video. Not being a mathematician, this one feels a good length down, but how can I tell...
@atreidesson
Жыл бұрын
But how did you use the characteristic of not 2?
@atreidesson
Жыл бұрын
@@freedomfilms974 well I suppose, doing nothing with it lies within that category
@iabervon
Жыл бұрын
He used it for the reverse direction at 4:44. The definition of a Lie algebra uses the condition that is stronger if char(F)=2, so a Lie algebra always has both properties, but you could define it with either if you only use fields with characteristic 0.
@SylComplexDimensional
11 ай бұрын
@@iabervon I had to watch this 1st vid to familiarize myself with Lie algebra (I’m BA, MSc Math) he’s claiming characteristic 2 is not zero & showing [x1,x1] = -[x1,x1] => [x1,x1] + [x1,x1] = 0
@SylComplexDimensional
11 ай бұрын
for all vectors x
@iabervon
11 ай бұрын
@@SylComplexDimensional If char(F) is not 2, then [x,x]+[x,x]=0 => [x,x]=0. If char(F)=2, then [x,x]+[x,x]=0 doesn't tell you anything, because y+y=0 for all y in a field of characteristic 2.
@scebsy6524
Жыл бұрын
is this inspired by the mathemaniac series?
@briansmith7458
11 ай бұрын
I am a hack.
@smb6995
Жыл бұрын
Nah bro you lieing! 😂
@krisbrandenberger544
Жыл бұрын
In the formula for the cross-product, the second term of the first component should be a_3*b_2, not a_3*b_1.
@synaestheziac
Жыл бұрын
So excited for this! I’ve been looking forward to it since you announced it months ago. Thanks Michael!
@robshaw2639
Жыл бұрын
In the derivation discussion of D(ab)... what does a product of the vectors a and b mean? The only thing we added to make the vector space a Lie algebra was the bracket...
@uptownfunk9446
Жыл бұрын
In that discussion, a and b are elements of A, which is not a Lie algebra but just an algebra, so you can multiply elements of A
@robshaw2639
Жыл бұрын
@@uptownfunk9446 So a general "algebra" has some kind of axioms on products of its elements? But then shouldn't a Lie algebra have those axioms, or is a Lie algebra not in fact an "algebra" in that sense?
@paulshin4649
Жыл бұрын
Might be wrong but I think an F-algebra is just an F-vector space A equipped with an F-bilinear map A×A->A called the *product* (usually denoted by concatenation). And a Lie F-algebra is an F-algebra whose product is alternating and satisfies the Jacobi identity, and we usually denote the product by [.,.] and call it the *Lie bracket*
@uptownfunk9446
Жыл бұрын
Yes, what @paulshin4649 wrote is correct. However it might be misleading, since "in the wild" many authors use the word algebra with implicit extra assumptions. The most common one is requiring the product to be associative, for instance, in which case you get an _associative algebra_, but which is sometimes also just called an algebra. Hence, a Lie algebra is an algebra in the most general sense as in the previous comment, but if someone requires an algebra to be associative then a Lie algebra will not anymore be necessarily an algebra in that sense. I hope that I wasn't too confusing!
@kristianwichmann9996
Жыл бұрын
Awesome. Would have loved to hear a bit more about Lie groups, though.
@Happy_Abe
Жыл бұрын
Very excited the math major videos are continuing!
@youtubepooppismo5284
Жыл бұрын
LOVE YOU
@robin1826
11 ай бұрын
Very excited to follow along in this series! For Exercise 3: Show [Eij,Ekl] = djk*Eil-dil*Ekj . I'm not sure how to proceed. I think I'm just struggling to parse the indices, what is this saying?
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