Root subsystems of rank 2 hyperbolic root systems

Abstract

Let $\Delta$ be a rank 2 hyperbolic root system. Then $\Delta$ has generalized Cartan matrix $H(a,b)= \left(\begin{smallmatrix} ~2 & -b\\ -a & ~2 \end{smallmatrix}\right)$ indexed by $a,b\in\mathbb{Z}$ with $ab\geq 5$. If $a\neq b$, 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 $a\neq b$, then $\Delta$ contains an infinite family of symmetric rank 2 hyperbolic root subsystems $H(k,k)$ for certain $k\geq 3$, 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 $a',b'\in\mathbb{Z}$ with $a'b'\geq 5$. One of our tools is a characterization of the types of root subsystems that are generated by a subset of roots. We classify these types of subsystems in rank 2 hyperbolic root systems.