Double Hall Algebras and Derived Equivalences

The resulting algebra is known as the Hall algebra HA of A. Hall algebras first appeared in the work of Steinitz [S] and Hall [H] in the case where A is the category of Abelian p-groups. They reemerged in the work of Ringel [R1]-[R3], who showed in [R1] that when A is the category of quiver representations of an A-D-E quiver ~ Q over a finite field Fq, the Hall algebra of A provides a realization of the nilpotent subalgebra Uq(n+) of the quantum group Uq(g) associated to the underlying graph of ~ Q. More generally, if ~ Q is of affine type, then the subalgebra of the Hall algebra generated by the simple objects corresponding to the vertices (known as the “composition subalgebra”) is isomorphic to the nilpotent subalgebra Uq(n+) of the quantum Kac-Moody algebra Uq(g) associated to ~ Q. In [R1], Ringel posed the question of how to extend this construction naturally to recover the whole quantum group Uq(g).Using the group algebra of the GrothendieckK0(A) to realize the torus algebra, he showed how to extend the Hall algebra in such a way that it recovers the Borel subalgebra Uq(b+) when A = RepFq( ~ Q). By generalizing the coproduct of Green [Gr] to this “extended” Hall algebra, Xiao [X2] showed that it is a self-dual Hopf algebra when the category A is hereditary (i.e. of homological dimension less than or equal to one). This is true in particular when A is the category of representations of a quiver. Using the Drinfeld double construction, Xiao defined an algebra DHA for a given hereditary category A that realizes the whole quantum group in the special case A = RepFq( ~ Q). It has remained an open question, however, whether this is the most natural way to realize Uq(g). Since the Hall