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Article overview
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The geometry of rank 2 hyperbolic root systems | | Lisa Carbone
; Scott H. Murray
; Sowmya Srinivasan
; | | Date: |
17 Jun 2015 | | Abstract: | Let $Delta$ be a rank 2 hyperbolic root system. Then $Delta$ has
generalized Cartan matrix $H(a,b)= left(egin{smallmatrix} ~2 & -b\ -a & ~2
end{smallmatrix}
ight)$ indexed by $a,binmathbb{Z}$ with $abgeq 5$. If
$a
eq b$, then $Delta$ is non-symmetric and is generated by one long simple
root and one short simple root, whereas if $a= b$, $Delta$ is symmetric and is
generated by two long simple roots. We prove that if $a$ and $b$ are both $>1$,
then no sum of real roots can be a real root. When $a$ or $b=1$, we classify
all the pairs of real roots whose sum is a real root. We prove that if $a
eq
b$, then $Delta$ contains an infinite family of symmetric rank 2 hyperbolic
root subsystems $H(k,k)$ for certain $kgeq 3$, generated by either two short
or two long simple roots. We also prove that $Delta$ contains non-symmetric
rank 2 hyperbolic root subsystems $H(a’,b’)$, for certain $a’,b’inmathbb{Z}$
with $a’b’geq 5$. | | Source: | arXiv, 1506.5405 | | Services: | Forum | Review | PDF | Favorites | |
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