We completely determine the structure constants between real root vectors in rank 2 Kac--Moody algebras $\mathfrak{g}$. 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 $S(a,b)=(\mathbb{Z}_{\geq 0}\alpha+\mathbb{Z}_{\geq 0}\beta)\cap \Delta^{\text{re}}(H(a,b))$. 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 $\ell$.

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2 阶 Kac-Moody 代数的换向器关系和结构常数

我们完全确定了 rank 2 Kac--Moody algebras $\mathfrak{g}$ 中实根向量之间的结构常数。我们的描述在计算上是有效的，即使在根格上的根系数随高度呈指数增长的 2 级双曲线情况下也是如此。我们的方法是将 Carter 寻找结构常数的方法从特殊对上的结构常数扩展到 2 级 Kac--Moody 情况。我们还确定了所有 2 阶 Kac-Moody 代数中仅涉及实根向量的所有交换子关系。$\mathfrak{g}$ 的广义 Cartan 矩阵的形式为 $H(a,b)= \left(\begin{smallmatrix} ~2 & -b\\ -a & ~2 \end{smallmatrix}\右)$ 其中 $a,b\in\mathbb{Z}$ 和 $ab\geq 4$。如果 $ab=4$，则 $\mathfrak{g}$ 是仿射类型。如果 $ab>4$，则 $\mathfrak{g}$ 是双曲线型。需要根串的显式知识，以及其和为实数的实根对的表征。我们证明，如果 $a$ 和 $b$ 都大于 1，那么任何实根之和都不能是实根。我们确定 $H(a,1)$, $a\geq 5$ 中实根 $\beta,\gamma$ 之间的根串，并确定集合 $S(a,b)=(\mathbb{Z} _{\geq 0}\alpha+\mathbb{Z}_{\geq 0}\beta)\cap \Delta^{\text{re}}(H(a,b))$。我们的工具之一是表征由根子集生成的根子系统。我们将这些子系统分类为 2 级 Kac--穆迪根系统。我们证明每个二阶无限根系统都包含一个无限族的非同构对称二阶双曲根子系统 $H(k,k)$ 对于某些 $k\geq 3$，由两个短根或两个长简单根生成.