Heisenberg doubles and derived categories.
Heisenberg doubles and derived categories.
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DOI：
10.1006/jabr.1997.7323
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发表时间：
1997-01
影响因子：
0.9
通讯作者：
M. Kapranov
中科院分区：
文献类型：
--
作者：
M. Kapranov
设A是同调维数为1的有限型阿贝尔范畴.利用绿色R（A）的结果，证明了A的广义Hall-Ringel代数具有自然的Hopf代数结构。我们考虑它的Heisenberg二重Heis（A），并研究它与A的导出范畴D（A）的关系。我们证明了Heis（A）可以看作是D^{0，1}（A）的一个Hall代数，D^{0，1}（A）是0度和1度复形的子范畴，在以下意义下：如果B是D（A）上位于D^{0，1}（A）中的一个t-结构的心，则R（B）自然是Heis（A）中的一个子代数.此外，我们定义了一个新的代数L（A），称为A的格代数，通过取R（A）的无限多个副本获得，一个用于无限一维格的每个站点，并在相邻站点的副本之间施加Heisenberg双型关系和在非相邻站点的副本之间施加Repeator型关系。这个代数在以下意义上充当全导范畴D（A）的“Hall代数”：任何导等价D（A）-->D（B）诱导格代数L（A）-->L（B）的同构。
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).