Positive Definite Cartan Matrices in Quantum Physics

[fresh_42]

TL;DR Summary

Cartan Matrices are not symmetric.

As I was looking for an example for a metric tensor that isn't among the usual suspects, I observed that the Cartan matrix I wanted to use is positive definite (I assume all are), but not symmetric. Are the symmetry breaks in quantum physics related to this fact?

 

[strangerep]

I observed that the Cartan matrix I wanted to use is positive definite (I assume all are), but not symmetric. Are the symmetry breaks in quantum physics related to this fact?

We'll need a bit more info on what you (think you) mean by "symmetry breaks in quantum physics". Spontaneous symmetry breaking? Pathologies at boundaries between different phases of matter? Something else?

 

[fresh_42]

I just wondered whether this asymmetry in the matrix that in the end defines a metric for mathematicians, and which I have been told years ago on PF is therefore essentially responsible that especially simple Lie groups play such a prominent role in QM, has any physical consequences and if, what they are. And please, don't open a distraction with $U(1)$ or the $A_l$ series.

 

[vanhees71]

What precisely are you talking about? What are "Cartan matrices"?

 

[fresh_42]

\begin{align*} A_l &: &\begin{pmatrix}2&-1&0&&& \cdots &&0 \\ -1&2&-1&0&&\cdots &&0\\ 0&-1&2&-1&0&\cdots&&0\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdots &\cdot&\cdot\\ 0&0&0&0&0&\cdots&-1&2 \end{pmatrix}\\ \\ \hline &&\\ B_l &: &\begin{pmatrix}2&-1&0&&\cdots&&&0\\ -1&2&-1&0&\cdots&&&0\\ \cdot&\cdot&\cdot&\cdot&\cdots&\cdot&\cdot&\cdot\\ 0&0&0&0&\cdots&-1&2&-2\\ 0&0&0&0&\cdots&0&-1&2 \end{pmatrix}\\ \\ \hline &&\\ C_l &: &\begin{pmatrix}2&-1&0&&\cdots&&&0\\ -1&2&-1&&\cdots&&&0\\ 0&-1&2&-1&\cdots&&&0\\ \cdot&\cdot&\cdot&\cdot&\cdots&\cdot&\cdot&\cdot\\ 0&0&0&0&\cdots&-1&2&-1\\ 0&0&0&0&\cdots&0&-2&2 \end{pmatrix}\\ \\ \hline &&\\ D_l &: &\begin{pmatrix}2&-1&0&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&0\\ -1&2&-1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&0\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&\cdot&\cdot&\cdot&-1&2&-1&0&0\\ 0&0&\cdot&\cdot&\cdot&\cdot&-1&2&-1&-1\\ 0&0&\cdot&\cdot&\cdot&\cdot&0&-1&2&0\\ 0&0&\cdot&\cdot&\cdot&\cdot&0&-1&0&2 \end{pmatrix}\\ \\ \hline && \end{align*}
\begin{align*} E_6 &: &\begin{pmatrix}2&0&-1&0&0&0\\ 0&2&0&-1&0&0\\ -1&0&2&-1&0&0\\ 0&-1&-1&2&-1&0\\ 0&0&0&-1&2&-1\\ 0&0&0&0&-1&2 \end{pmatrix}\\ \\ \hline &&\\ E_7 &: &\begin{pmatrix}2&0&-1&0&0&0&0\\ 0&2&0&-1&0&0&0\\ -1&0&2&-1&0&0&0\\ 0&-1&-1&2&-1&0&0\\ 0&0&0&-1&2&-1&0\\ 0&0&0&0&-1&2&-1\\ 0&0&0&0&0&-1&2 \end{pmatrix}\\ \\ \hline &&\\ E_8 &: &\begin{pmatrix}2&0&-1&0&0&0&0&0\\ 0&2&0&-1&0&0&0&0\\ -1&0&2&-1&0&0&0&0\\ 0&-1&-1&2&-1&0&0&0\\ 0&0&0&-1&2&-1&0&0\\ 0&0&0&0&-1&2&-1&0\\ 0&0&0&0&0&-1&2&-1\\ 0&0&0&0&0&0&-1&2 \end{pmatrix}\\ \\ \hline\\ F_4 &: &\begin{pmatrix}2&-1&0&0\\ -1&2&-2&0\\ 0&-1&2&-1\\ 0&0&-1&2 \end{pmatrix}\\ \\ \hline &&\\ G_2 &: &\begin{pmatrix}2&-1\\ -3&2 \end{pmatrix}\\ \\ \hline && \end{align*}

 

[Vanadium 50]

Someone is good with TeX!

 

[fresh_42]

Someone is good with TeX!

Well, yes, but this time Ctrl+V was sufficient:
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-structures/