AR quivers, exceptional sequences and algorithms in derived Hall algebras

Consider the canonical isomorphism between the positive part U ⁺ of the quantum group U _q (g) and the Hall algebra H(Λ), where the semisimple Lie algebra g and the finite-dimensional hereditary algebra Λ share a Dynkin diagram. Chen and Xiao have given two algorithms to decompose the root vectors into linear combinations of monomials of Chevalley generators of U ⁺, respectively induced by the braid group action on the exceptional sequences of Λ-modules and the structure of the Auslander-Reiten quiver of Λ. In this paper, we obtain the corresponding algorithms for the derived Hall algebra DH(Λ), which was introduced by Toën. We show that both algorithms are applicable to the lattice algebra and Heisenberg double in the sense of Kapranov. All the new recursive formulae have the same flavor with the quantum Serre relations.