Derived Hall algebras of root categories

For a finitary hereditary abelian category $\mathcal{A}$, we define a derived Hall algebra of its root category by counting the triangles and using the octahedral axiom, which is proved to be isomorphic to the Drinfeld double of Hall algebra of $\mathcal{A}$. When applied to finite-dimensional nilpotent representations of the Jordan quiver or coherent sheaves over elliptic curves, these algebras provide categorical realizations of the Drinfeld double of the ring of symmetric functions and also double affine Hecke algebras.