What are...octonions?

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Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.
This time.
What are...octonions? Or: Division = good, associativity = bad.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Slides.
www.dtubbenhaue...
TeX files for the presentation.
github.com/dtu...
Thumbnail.
en.wikipedia.o...
Main discussion.
www.taylorfran...
link.springer....
www.ams.org/jo...
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golem.ph.utexa...
Background material.
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Computer talk.
magma.maths.us...
doc.sagemath.o...
mathematica.st...
Pictures used.
useruploads.so...
cdn1.byjus.com...
Picture from simple.wikiped...
en.wikipedia.o...
thatsmaths.fil...
johncarlosbaez...
en.wikipedia.o...
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RU-vid and co.
• JOHN BAEZ | SPLIT OCTO...
• Talk 8: Can We Underst...
• This Week's Finds 9: q...
#algebra
#representationtheory
#mathematics

Опубликовано:

30 сен 2024

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Комментарии : 15

@fuseteam 2 месяца назад

_Math has the Reals_ Double or nothing? Mathematicians: let's go! _Math now has the Complex_ Double or nothing? Mathematicians: let's go!! _Math now has Quaternions_ Double or nothing? Mathematicians: let's go!!! _Math now has Octonions_ Double or nothing? Mathematicians: let's go!!!! _Sedenions entered the chat_ Oh Shi- go back!

@VisualMath 2 месяца назад

Haha, so accurate 😀

@SummiyaMumtaz Месяц назад

If i have h=(1+10i+10j+10k+10l+10m+10n+10o) then what is h^2?

@VisualMath Месяц назад

Try (copy-paste) the below code in the Magma online calculator magma.maths.usyd.edu.au/calc/: > fano := {@ : n in [0..6] @}; > T := [ : g in Sym(3), f in fano]; > T cat:= [ : i in [2..8] ]; > T cat:= [ : i in [1..8] ] cat [ : i in [2..8] ]; > octonions := func< R | Algebra< R, 8 | T > >; > OZ := octonions(Integers()); > PA := PolynomialAlgebra(Integers(),8); > h:=e1+10*e2+10*e3+10*e4+10*e5+10*e6+10*e7+10*e8;

@Jaylooker 10 месяцев назад

The Cayley-Dickson construction relates R, C, H, O, and S of the reals, complex numbers, quaternions, octionons, and sedenions as described in this video. It has many applications. At least in topology it can be use to derive the four Hopf fibrations and Bott periodicity. Notably, Bott periodicity is the basis for stable homotopy theory and K-theory. See “The Octonions” (2001) by John Baez or your AMS link in the description.

@VisualMath 10 месяцев назад

Thanks for the link, the good old pi_k(orthogonal groups). Imho Bott periodicity is somewhat difficult to explain without throwing too much background at the viewer, so I went with something easier in the next video as an application of octonions: Fruendenthal's magic square 😀

@Jaylooker 10 месяцев назад

@@VisualMath Yup. There is also π(U(n)) for the unitary groups U(n). Bott periodicity is difficult to explain and it applies to all classical Lie groups. The Freudenthal magic square does relate most of the exceptional Lie algebras.

@eternaldoorman5228 10 месяцев назад

@@VisualMathWas there really a man called Jacques Tits? Sometimes you have to wonder about these things, ...

@VisualMath 10 месяцев назад

@@eternaldoorman5228 I know someone who has a joint paper with Tits, so I go for "real" 😁

@yankeed4793 10 месяцев назад

Again, a nice quick intro to a complex topic which motivates interest in a deeper dive, even for a “layperson”. I’d long wondered if octonions were the foundation of the “eightfold way” which figured prominently in the early history of quantum field theory - and so they are. You emphasize the importance of invertability, which is critical for a group property and thereby unlocking the tools for analyzing the symmetries of the standard model through group theory, explaining the connection. Murray Gell-Mann, of “quark” fame, later lamented that his lunch group in the university cafeteria included one of leading mathematicians in the field of Lie Algebras and he could have saved himself enormous time if he had paid closer attention to this since he ended up reinventing much of this work on his own. Looking forward to the next video where you discuss further!

@VisualMath 10 месяцев назад

Excellent, thanks for pointing out the standard model! I learned that the octonions O are used from this fabulous resource here: math.ucr.edu/home/baez/standard/ I am also very glad that you like the story, same here 😀 There are many aspects of O that I cannot touch upon (like the standard model). But, historically backwards 😅, I will try to explain in the follow-up how the exceptional Lie algebras (so exceptional continuous symmetries) arise from O.

@eternaldoorman5228 10 месяцев назад

4:52 this is beautiful! Since you get negative numbers by multiplying by i^2 I wonder if there is a way to construct all the number systems "top down", starting with some very primitive algebraic structure like a magma. I need to play with this doubling operator a bit more to find out what I'm talking about 😂.

@VisualMath 10 месяцев назад

The doubling is amazing, isn't it ☺ I am not sure what you have in mind, but I am curious. Let us know if you make any progress 😀

@eternaldoorman5228 9 месяцев назад

@@VisualMath I can't say I've thought about this enough to be able to say more about my vague idea. Another question I have is whether is some class of equations that quaternions and octonions allow you to solve, I mean, negative numbers allow you to solve equations like x + a = 0 complex numbers allow you to solve equations like x^2 + a = 0, can you solve equations like x - (x + 1) = 0 or something with quaternions? I ask because I did once manage to 'solve' x = x + 1 using pairs of complex numbers in some sort of conjugate relationship. I can't remember the details however because I lost my computer and the backups and all my notes about it.

@VisualMath 9 месяцев назад

@@eternaldoorman5228 Very interesting. I have never seen solutions to x=x+1. I feel like Octonions might not be helpful here, but I am not 100% sure. The only thing I remember is that polynomials have infinite many solutions in Octonions. So you get many more roots. New solutions might also show up if you go to more sophisticated equations, like involving analytic functions. But I am not sure about the state of the arts here.