Root vectors arising from Auslander-Reiten quivers (Q1969487)

Let \(\mathfrak g\) be a semisimple Lie algebra, and \(U_q({\mathfrak g})\) its quantum enveloping algebra. Let \(\Lambda\) be the associated finite dimensional hereditary algebra over a finite field \(k\). In this paper the authors obtain a new algorithm to decompose the root vectors corresponding to preprojective and preinjective indecomposable \(\Lambda\)-modules into a linear combination of monomials of the canonical generators of \(U_q^+({\mathfrak g})\), the positive part of \(U_q({\mathfrak g})\). This algorithm depends only on the structure of the AR-quiver of \(\Lambda\). In particular, the algorithm is complete for all root vectors in any type case, i.e., any case of \(A_n\), \(B_n\), \(C_n\), \(D_n\), \(E_i\) (\(i=6,7,8\)), \(F_4\), and \(G_2\).