dorsal/arxiv
View SchemaHeisenberg doubles and derived categories
| Authors | M. Kapranov |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9701009 |
| URL | https://arxiv.org/abs/q-alg/9701009 |
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).
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"abstract": "Let A be an abelian category of finite type and homological dimension 1. Then\nby results of Green R(A), the extended Hall-Ringel algebra of A, has a natural\nHopf algebra structure. We consider its Heisenberg double Heis(A) and study its\nrelation with D(A), the derived category of A. We show that Heis(A) can be\nviewed as a \"Hall algebra\" of D^{0,1}(A), the subcategory of complexes situated\nin degrees 0 and 1, in the following sense: if B is the heart of a t-structure\non D(A) lying in D^{0,1}(A), then R(B) is naturally a subalgebra in Heis(A).\nFurther, we define a new algebra L(A) called the lattice algebra of A, obtained\nby taking infinitely many copies of R(A), one for each site of an infinite\n1-dimensional lattice and imposing Heisenberg double-type relations between\ncopies at adjacent sites and oscillator-type relations between copies at\nnon-adjacent sites. This algebra serves as the \"Hall algebra\" of the full\nderived category D(A) in the following sense: any derived equivalence\nD(A)--\u003eD(B) induces an isomorphism of lattice algebras L(A)--\u003eL(B).",
"arxiv_id": "q-alg/9701009",
"authors": [
"M. Kapranov"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Heisenberg doubles and derived categories",
"url": "https://arxiv.org/abs/q-alg/9701009"
},
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