Bridgeland’s Hall algebras and Heisenberg doubles

Abstract

Let [Formula: see text] be a finite-dimensional hereditary algebra over a finite field, and let [Formula: see text] be a fixed integer such that [Formula: see text] or [Formula: see text]. In the present paper, we first define an algebra [Formula: see text] associated to [Formula: see text], called the [Formula: see text]-periodic lattice algebra of [Formula: see text], and then prove that it is isomorphic to Bridgeland’s Hall algebra [Formula: see text] of [Formula: see text]-cyclic complexes over projective [Formula: see text]-modules. Moreover, we show that there is an embedding of the Heisenberg double Hall algebra of [Formula: see text] into [Formula: see text].