Root Vectors Arising from Auslander–Reiten Quivers,

Consider the canonical isomorphism between the Ringel–Hall algebra (Λ) and the positive part U + of the Drinfield–Jimbo quantum group Uq(g), where the finite dimensional hereditary algebra Λ and the semisimple Lie algebra g enjoy a common Dynkin diagram. We get an algorithm to decompose the root vectors into combinations of monomials of the canonical generators Ei in the quantum group depending only on the structure of the Auslander–Reiten quiver of Λ. The algorithm is deduced by using a similar method as we did in X. Chen and J. Xiao [1999, Compositio Math.117, No. 2, 161–187], particularly, the derivation defined there.