Central polynumber algebra and a (baby) Weyl character formula | Math Founds 236 | N J Wildberger
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- čas přidán 27. 07. 2024
- The multiset approach to arithmetic that we are here developing can be applied to re-interpret and deepen our understanding of the 20th century's most important formula: the Weyl character formula for the (irreducible) representations of simple Lie groups and Lie algebras.
The special cases that we are going to describe connect with the two most basic ADE graphs, the graphs A1 and A2, corresponding to the representations of SU(2) and SU(3), which coincidently are arguably the two most important non-commutative Lie groups in modern particle physics and feature centrally in the Standard Model. The group SU(2) is also the three dimensional sphere which occurs in the four dimensional algebra of quaternions, which we quickly review.
We also briefly mention the great 20th century mathematician Hermann Weyl.
The Weyl character formula viewed correctly is a statement about polynumbers, not complex exponentials as found in almost all texts and papers on the subject. Irrationalities must be avoided if we are going to understand mathematics, and physics, properly!
Correction at 20:59 I mention the classsification of simple Lie algebras, and forget to include the other two families, of types B_n and C_n.
The tensor product of representations corresponds in some fashion to interactions between elementary particles, and for this the characters of the representations are very useful, as the questions reduce to computing ordinary products of these. We illustrate this in the SU(2) case of central polynumbers, showing how the Weyl character formula in this situation connections with tensor product multiplicities and can be computed using our integral polynumber algebra.
This lecture is also closely connected to some of the topics that we discuss in the Exceptional Structures in Mathematics and Physics via Dynamics on Graphs exploration series available to Members of the Wild Egg Maths channel, and also to Patreon supporters. In fact I will develop the contents of this lecture in more detail and in greater generality in that series.
Video Contents:
00:00 Central polynumbers and relations
4:02 A1 Weyl character formula
7:27 Examples of the A1 WCF
9:03 Quaternions
15:05 Hermann Weyl
20:44 The rational Weyl character formula
30:12 Corresponding characters
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This is off topic but a geometric algebra course by Wildberger would be incredible!
Yes! Please!
Thank you for all youTube free video s
This is an interesting way of using polynumbers, instead of exponentials.
In Humphrey's Lie Algebras book, he has a Weyl character formula with no exponentials
Glad to see that you are still making videos Norman. You have a perfect way of teaching. Thank You.
I have no idea what you're talking about, but I understood every word. If you think that's paradoxical, well call me the barber who shaves anyone who doesn't shave himself.
Very insightful and enlightening. I will look specifically using this insight at the Lie Groups A3 D5[E5] E6 E7 E8 F4 G2, because this finite pattern of 1 Lie Group A3 + 6 Lie Group families have special qualities maybe related to the cosmic and particle zoo via the Sporadic Finite Simple Group Framework.
Love this series
Fascinating lecture. The approach of using grids as intuition for multinumber arithmetic is something which really piques my interest. After seeing this practice on the channel a couple of years ago, I tried using it (and some previous thought I'd put in that direction) to generalize aspects of positional notation into two dimensions. Unfortunately, I don't think I got very far. Typically important examples (at least as algebraic curves) didn't pan out, and some weird behavior I found in one dimension didn't seem to generalize. Perhaps it's because I was treating reducibility as the enemy while keeping in mind the constraints I'd laid down.
I have noticed a deep connection between the mapping of S^3 to 3D visualizations on one hand, and the photographs that arise from particle collisions. The coincidence isn't surprising, the Standard Model is built on the gluons base which is represented by SU(2) which is equivalent to S^3
We can see a version of "mysterious quantum shift" here. Interpreting < and > as mark and antimark intergral elements of the numerator and and as the denominator element of mereological fractions and interpreting the words as multisets, we get from outwards seed < > the standard numerical structure of Stern-Brocot type structure.
< >
< >
< >
etc., with numerical interpretation 1:0 1:1 0:1 1:1 1:0 of the 3rd row.
What about the inwards seed, when applying the same procedure?
> <
> >< <
> >>< >< > >>>< >>< >>< >< >>< has 3>0and1 as the mark-antimark structure of integers, we can do simple negation by canceling > and < pairs from the words, when they are not already part of the denominator element . After 'Del>
Bro what in the world was that?? It’s just the half sum of the positive restricted roots in Lie algebra theory, idk why you physics ppl call it a mysterious “quantum shift” or what any of that above is….
Thankyou
Here is a recent video on the Weyl Character Formula presented by Nigel Higson.
czcams.com/video/BoNFX46OtBE/video.html
Thanks!
@14:32 You say the circle itself in a compact Lie group but "it is not abelian" but this SU(2) of the 3 sphere is the smallest non-abelian compact Lie group. What is the difference between NOT abelian of the circle and and NON abelian of the sphere. Did you mean to say that the circle S1 is "not non abelian"?
Hi John, Yes I misspoke, the circle is clearly abelian.
Really nice videos sir, but I noticed an error. S^3 is not the unit sphere, that is S^2,
I believe S^3 is not a 3-dimensional space like the sphere you visualize, the 2-sphere is that (S^2). They are important distinctions. S^3 is wildly different. S refers to the surface we embed in R which is a dimension higher. That point, and more of S^3, would be really interesting to get into full detail, I don't think it is trivial.
S^3 is the three dimensional sphere in four dimensional space. It is a Lie group, and just another manifestation of SU(2)
Do you think that in reality this S^3 group action could still exist in our universe (even though that appears spatially 3 dimensions) , or can we (directly) relate S^3 to 4d space-time? I don't think it is coincidence it is the same representation of the strong force!!!
Or that this spherical geometry (S^3) is also admitted by the universe, in contrast to only thinking about the universe as Euclidean or R^3 ?
@@njwildberger I know you're not a physicist sir, but I am extremely interested in hearing your thoughts on these questions. Sorry if you didn't have the time to reply yet