Heisenberg doubles and derived categories
| dc.creator | Kapranov, M. | |
| dc.date | 1997-01-09 | |
| dc.date.accessioned | 2026-07-07T09:17:25Z | |
| dc.date.available | 2026-07-07T09:17:25Z | |
| dc.description | 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). | |
| dc.description | 30 pages, plain TEX | |
| dc.identifier | https://arxiv.org/abs/q-alg/9701009 | |
| dc.identifier | http://arxiv.org/abs/q-alg/9701009 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/153658 | |
| dc.subject | Quantum Algebra | |
| dc.title | Heisenberg doubles and derived categories | |
| dc.type | text |