How Did Lie Groups Become Central to Physics?

[fresh_42]

I explain by simple examples (one-parameter Lie groups), partly in the original language, and along the historical papers of Sophus Lie, Abraham Cohen, and Emmy Noether how Lie groups became a central topic in physics. Physics, in contrast to mathematics, didn’t experience the Bourbakian transition so the language of for example differential geometry didn’t change quite as much during the last hundred years as it did in mathematics. This also means that mathematics at that time has been written in a way that is far closer to the language of physics, and those papers are not as old-fashioned as you might expect.

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[Office_Shredder]

This is great, thank you.

 

[strangerep]

Under the section "Invariants":$$U.f =\xi \dfrac{\partial f}{\partial x}+\eta\dfrac{\partial f}{\partial x}\equiv 0.$$Should the 2nd term be a $\partial/\partial y$ ?

 

[malawi_glenn]

In the intro "QED" should be "SM (the standard model)"

 

[Couchyam]

It might also help to explain that the focus in math shifted to algebraic geometry and more general algebraic groups. Also, I think there might still be at least some finite, nonzero level of mathematical interest in the mysterious and unexpected connection between semi-simple Lie groups (or even just finite subgroups of SU(2)) and Dynkin diagrams.

 

[fresh_42]

It might also help to explain that the focus in math shifted to algebraic geometry and more general algebraic groups.

I have a fancy book about buildings, but I'm afraid we won't have enough readers for an article about Coxeter groups.

Also, I think there might still be at least some finite, nonzero level of mathematical interest in the mysterious and unexpected connection between semi-simple Lie groups (or even just finite subgroups of SU(2)) ...

Is there another finite subgroup besides $\{\pm 1\}$?

... and Dynkin diagrams.

https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-structures/#7-Dynkin-Diagrams

 

[Couchyam]

Is there another finite subgroup besides $\{\pm 1\}$?

Well, there are the cyclic groups of order $N$, dihedral groups, and the symmetries of Platonic solids, which I think McKay noticed correspond (tantalizingly) to the (possibly extended) A, D, and E Dynkin diagrams respectively.
https://en.wikipedia.org/wiki/McKay_graph

 

[martinbn]

Is there another finite subgroup besides $\{\pm 1\}$?

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC391682/pdf/pnas00617-0279.pdf

 

[Couchyam]

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC391682/pdf/pnas00617-0279.pdf

Exactly.

 

[mathwonk]

@Couchyam: one tiny remark: the dihedral groups apparently do not lie in SU(2), since SU(2) has only one element of order 2; rather it is the dicyclic groups which do. So apparently the problem is the absence of reflections. This seems to also rule out the (full) symmetry groups of the Platonic solids. In general, as you no doubt know, one uses the double cover of SO(3) by SU(2) to pull back finite rotation groups, thus getting certain double covers of the rotation groups of the Platonic solids ("binary icosahedral", "binary octahedral"...), rather than their full symmetry groups. I am far from expert here, have not done the calculations, and am taking what I read somewhat on faith, but it sounds right.
https://math.stackexchange.com/questions/40351/what-are-the-finite-subgroups-of-su-2c