Commutator relations and structure constants for rank 2 Kac--Moody algebras

TL;DR

This paper provides a comprehensive and efficient method to determine structure constants and commutator relations for real root vectors in rank 2 Kac--Moody algebras, including hyperbolic cases with exponential root growth.

Contribution

It extends Carter's method to rank 2 Kac--Moody algebras, classifies root subsystems, and explicitly describes structure constants and root strings, especially in hyperbolic cases.

Findings

Complete determination of structure constants between real root vectors.

Classification of rank 2 root subsystems, including hyperbolic types.

Proof that certain root sums are not real roots when both parameters exceed one.

Abstract

We completely determine the structure constants between real root vectors in a rank 2 Kac--Moody algebra $\mathfrak{g}$. Our description is computationally efficient, even in the rank 2 hyperbolic case where the coefficients of roots on the root lattice grow exponentially with height. Our approach is to extend Carter's method of finding structure constants from those on extraspecial pairs to the rank 2 Kac--Moody case. We also determine all commutator relations involving only real root vectors in all rank 2 Kac-Moody algebras. The generalized Cartan matrix of $\mathfrak{g}$ is of the form $H(a,b)= \left(\begin{smallmatrix} ~2 & -b\\ -a & ~2 \end{smallmatrix}\right)$ where $a,b\in\mathbb{Z}$ and $ab\geq 4$. If $ab=4$, then $\mathfrak{g}$ is of affine type. If $ab>4$, then $\mathfrak{g}$ is of hyperbolic type. Explicit knowledge of the root strings is needed, as well as a characterization of the pairs of real roots whose sums are real. We prove that if $a$ and $b$ are both greater than one, then no sum of real roots can be a real root. We determine the root strings between real roots $\beta,\gamma$ in $H(a,1)$, $a\geq 5$ and we determine the sets $(\mathbb{Z}_{\geq 0}\alpha+\mathbb{Z}_{\geq 0}\beta)\cap\Delta^{\text{re}}(H(a,b))$. One of our tools is a characterization of the root subsystems generated by a subset of roots. We classify these subsystems in rank 2 Kac--Moody root systems. We prove that every rank two infinite root system contains an infinite family of non-isomorphic 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 a non-symmetric hyperbolic root systems $H(a,b)$ with $a\ne b$ and $ab>5$ also contains an infinite family of non-isomorphic non-symmetric rank 2 hyperbolic root subsystems $H(a\ell,b\ell)$, for certain positive integers $\ell$.