Coxeter transformations in quantum groups (Q5960130)

Let \(\Delta=(I,(\;,\;))\) be a Cartan datum of finite type, \(U_q(\mathfrak g)\) the quantized enveloping algebra of type \(\Delta\), \(h\) the Coxeter number of \(\Delta\). It is natural to define a Coxeter transformation on \(U_q(\mathfrak g)\) as \(T=T_{i_1}T_{i_2}\cdots T_{i_n}\), where \(\{i_1, i_2, \dots, i_n\}\) is an arrangement of \(I\). In this paper, the authors show that any two Coxeter transformations on \(U_q(\mathfrak g)\) are conjugate if the diagram \(\Delta-(I,(\;,\;))\) is a tree. Furthermore, by the Ringel-Hall algebra approach, they show that \[ T^h(x_{\alpha})=v^{-(\alpha,\alpha)}x_{\alpha}K_{-2\alpha}, \] for any weight vector \(x_{\alpha}\in U_q(\mathfrak g)\) with weight \(\alpha\in \mathbb Z[t]\).