Realization of ıquantum groups via Δ-Hall algebras

Abstract

For an essentially small hereditary abelian category A, we associate an algebra HΔ(A), called the Δ-Hall algebra of A. The basis of HΔ(A) is the isomorphism classes of objects in A, and the Δ-Hall numbers calculate certain three-cycles of exact sequences in A. We show that the Δ-Hall algebra HΔ(A) is isomorphic to the 1-periodic derived Hall algebra associated to A. By taking suitable extension and twisting, we can obtain the ıHall algebra and the semi-derived Hall algebra related to A respectively.When applied to the nilpotent representation category A=repnil(kQ) for an arbitrary quiver Q without loops, the (resp. extended) Δ-Hall algebra provides a realization of the (resp. universal) ıquantum group associated to Q.