Root strings with three or four real roots in Kac-Moody root systems
The author gives a necessary and sufficient condition, in terms of generalized Cartan matrices, for a Kac-Moody root system to have a root string with three or four real roots. Let \(A=(a_{ij})\) be an \(n\times n\) generalized Cartan matrix, \(\Delta\) the associated root system, and \(\Delta^{re}\) the set of real roots. Put \(r(\alpha;\beta)=\#| \{\beta +k\alpha | \quad k\in {\mathbb{Z}}\}\cap \Delta^{re}|\) for \((\alpha,\beta)\in \Delta^{re}\times \Delta.\) Then the following two conditions are equivalent. (1) \(r(\alpha;\beta)=3\) or 4 for some \((\alpha,\beta)\in \Delta^{re}\times \Delta\). \((2)\quad a_{ij}=- 1\quad and\quad a_{ji}<-1\) for some i,j (1\(\leq i,j\leq n).\) As a corollary, one can simplify the Steinberg-Tits presentation of the associated Kac-Moody group in the case when A has a certain property.
scientific article; zbMATH DE number 4071148