Lie algebras with @TomRocksMaths

Teaching Tom Crawford a bit about my favorite subject -- Lie algebras.

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[TomRocksMaths]

Thanks for having me - I learned a lot that's for sure!

[dmytro_shum]

No, stop, that's enough. Too many lies. I can't stand it! )))

[ronwittmann]

This collaboration is helpful. As a physicist, I have been exposed to Lie algebras and would like to know more. However, the more general, rigorous treatments presented by many mathematicians are almost unrecognizable to me.

[ConManAU]

This was a fantastic pair of videos! I love how the two topics were intertwined with each other, and how we got to share in the “aha” moments with the respective “students”.

I’m a fan of Socratic learning, so I particularly liked when the student got to contribute to the next step of the proof, and I’d love to see more of that happen if you do more of these videos.

[thehappyapy]

This was a delightful presentation! I love everything with Lie algebras. It was great seeing a Heisenberg Lie algebra, as that is not common in introductions. I also really appreciated the multiple nods towards representation theory and the lattice of weights.

[bastianfrom]

I remember as a student in the 2nd or 3rd semester I came across SL2 and suddenly thought - hey that looks like x-Product. I worked out all the details over night, looked in 2-3 books about linear algebra which where available at that time in order to find out if this was known and I couldn't find it. Then I went to my math professor who still is quite a luminary in Lie-Algebras (he wrote a couple of books about that subject) and I found out this was a long known connection :( . You just reminded me of that sad epsiode. But anyway I still think, that was not too bad for a physics student and it was quite fun working it out by myself.

[scipionedelferro]

What a treat!! I'm a physicist and I love both channels, seeing you guys together on one of my favorite topic (Lie algebra) is incredible! I hope you can make more of these cross-over!

[CM_France]

Hi,

I just got that we are taking about polynomial transformations and not polynomials themselves.

Very nice teamwork and application of the Heisenberg's Uncertainty Principle !

[NeilGirdhar]

8 minutes in and I'm already really enjoying this.

[shuewingtam]

Great job Thanks but i found that basis vector k x j should be equal to -i not i. Is that right?

[gavintillman]

Really looking forward to this. Will watch later tonight. Just watched the QM one which I enjoyed. Interesting pairing as I did the QM in my second year at Cambridge, whereas I didn’t get an opportunity to do Lie Algebras at undergrad level. I think some years it was available in Part III but not my year. Shame, as finite simple groups were a big interest of mine and Lie Algebras is very complementary to that. The QM I studied in 1985, way before the internet, so my choices of presentation were either a not great lecturer, or some dry textbooks. So envious of today’s students who can look at several different order presentations of a subject on YouTube, such as Binney or Susskind. Ironically, I reckon I understand QM now, despite leaving academia in 1988, than I did at the time, because of this! Keep up the good work, I follow both channels closely.

[Andrew-rcvh]

Edgar F. Codd invented an algebra to do with database tables. They have inner joins and outer joins and they also use a cross product join sometimes. That is every record to record combination of the two tables.

[misterlau]

Alright, but I'm enjoying your class because I'm kinda old and I already know these maths, but not much resources like this back then...

Good job

[drdca]

An associative algebra is just a ring homomorphism (possibly where the domain is required to be a field, but this isn’t super-necessary)

[jantarantowicz]

Oh man, I love how your channel grew. I loved and love your videos about competition problems, however i really LOVE that you trying to introduce some higher mathematics to us. I like to think that you might be making your viewers feel that theoretical mathematics isn't that scary and complicated and I'm here for that! <3

[FleuveAlphee]

Just noting that, in the UK, the cross product is introduced to college (Sixth Form) students at pre-U level, as part of A level 'Further Maths' (currently FP1 in the post-2017 syllabus).

[dickinaround]

It was nice to see more of your personality in between the math bits - a nice shake up from the refined, concise presentations you typically present.

[SimonJackson]

"The zero is a number with an information destructive projectaform sub set morphism. ..." :D

[trumanburbank]

Maybe we can make that dancing AxA vector fill in a planer disk rather than getting cancelled out. So, the cross product in three dimensions is orthogonal to the two vectors. We could consider a vector as a disk with an infinite ray on one side, and the magnitude of the vector as the radius of the disk. Then the cross product would produce another ray along the intersection of the two disks (using the right-hand rule), and the radius of the resultant disk would be ||A|| ||B||. Now when we perform AxA, that which intersects are the entire disks, and the rays themselves. So, the disk is still there with a radius of ||A||^2, with the 'new' ray, the same as A. Still one problem though -- what do we do with (-A)xA ? It seems there is some ambiguity as to which way the ray would point. I think we could replace the word 'radius' with 'area' and get a workable analysis either way. I have a feeling that there is something in Geometric Algebra that goes into that(?)

[KusacUK]

Bit surprised you both feel that you don’t get introduced to dot and cross products until university/college. I had to do it as part of A Levels, and using quaternions, so we were well aware of the loss of associativity. Admittedly there was none of the formality around fields/rings and algebras, was more around having tools to solve particular problems in vector space.