Heisenberg doubles and derived categories

Abstract

Let A be an abelian category of finite type and homological dimension 1. Then by results of Green R(A), the extended Hall-Ringel algebra of A, has a natural Hopf algebra structure. We consider its Heisenberg double Heis(A) and study its relation with D(A), the derived category of A. We show that Heis(A) can be viewed as a "Hall algebra" of D^{0,1}(A), the subcategory of complexes situated in degrees 0 and 1, in the following sense: if B is the heart of a t-structure on D(A) lying in D^{0,1}(A), then R(B) is naturally a subalgebra in Heis(A). Further, we define a new algebra L(A) called the lattice algebra of A, obtained by taking infinitely many copies of R(A), one for each site of an infinite 1-dimensional lattice and imposing Heisenberg double-type relations between copies at adjacent sites and oscillator-type relations between copies at non-adjacent sites. This algebra serves as the "Hall algebra" of the full derived category D(A) in the following sense: any derived equivalence D(A)-->D(B) induces an isomorphism of lattice algebras L(A)-->L(B).