University of IsfahanTransactions on Combinatorics2251-86578420191201Elliptic root systems of type $A_1$, a combinatorial study11212402310.22108/toc.2019.117338.1648ENZahraKharaghaniDepartment of mathematics, University of Isfahan, Isfahan, IranJournal Article20190529We consider some combinatorics of elliptic root systems of type $A_1$. In particular, with respect to a fixed reflectable base, we give a precise description of the positive roots in terms of a ``positivity'' theorem. Also the set of reduced words of the corresponding Weyl group is precisely described. These then lead to a new characterization of the core of the corresponding Lie algebra, namely we show that the core is generated by positive root spaces.https://toc.ui.ac.ir/article_24023_773c7ef2a4751d9ad0477a8e0cf7b337.pdf