论文标题
霍尔代数的确切结构和变性
Exact structures and degeneration of Hall algebras
论文作者
论文摘要
我们研究了由程度函数在不可分解的对象的同构类别中引起的确切类别的霍尔代数的退化。我们证明，大厅代数$ \ Mathcal {h}（\ Mathcal {e}）$的每一次这种变性的确切类别$ \ Mathcal {e} $是较小精确结构$ \ MATHCAL {e}'<\ nathCal {e and and and aDdivese $ wher a <\ \ naterive a'<\ naterive a''<\ \ naterive a'<\ naterive n \ n a <\ \ nathers aDdive wher n \ n时$ \ MATHCAL {E} $在Inomoto的意义上是可以接受的，对于任何$ \ Mathcal {e}'<\ Mathcal {e} $满足适当的有限条件，我们证明$ \ MATHCAL { $ \ Mathcal {h}（\ Mathcal {e}）$。 在加上有限的情况下，所有此类函数构成了一个简单锥的圆锥体，其脸部晶格反映了精确结构的晶格的性质。对于Dynkin Quivers的表示类别，我们恢复了相应量子组负部分的退化，以及由Fourier，Reineke和第一作者研究的相关的多面体结构。 在此过程中，我们对Inomoto和Brüstle-Langford-Hassoun-Roy的某些结果进行了较小的改进，内容涉及添加剂类别的确切结构的分类。我们证明，对于每个基于完整的添加剂类别$ \ MATHCAL {a} $，存在一个Abelian类别，其Serre子类别的晶格与$ \ Mathcal {a} $上的精确结构的晶格是同构的。我们表明，每个Krull-Schmidt类别都承认一个独特的最大可允许的确切结构，并且可允许的精确结构的较小精确结构的晶格是布尔的。
We study degenerations of the Hall algebras of exact categories induced by degree functions on the set of isomorphism classes of indecomposable objects. We prove that each such degeneration of the Hall algebra $\mathcal{H}(\mathcal{E})$ of an exact category $\mathcal{E}$ is the Hall algebra of a smaller exact structure $\mathcal{E}' < \mathcal{E}$ on the same additive category $\mathcal{A}.$ When $\mathcal{E}$ is admissible in the sense of Enomoto, for any $\mathcal{E}' < \mathcal{E}$ satisfying suitable finiteness conditions, we prove that $\mathcal{H}(\mathcal{E}')$ is a degeneration of $\mathcal{H}(\mathcal{E})$ of this kind. In the additively finite case, all such degree functions form a simplicial cone whose face lattice reflects properties of the lattice of exact structures. For the categories of representations of Dynkin quivers, we recover degenerations of the negative part of the corresponding quantum group, as well as the associated polyhedral structure studied by Fourier, Reineke and the first author. Along the way, we give minor improvements to certain results of Enomoto and Brüstle-Langford-Hassoun-Roy concerning the classification of exact structures on an additive category. We prove that for each idempotent complete additive category $\mathcal{A}$, there exists an abelian category whose lattice of Serre subcategories is isomorphic to the lattice of exact structures on $\mathcal{A}$. We show that every Krull-Schmidt category admits a unique maximal admissible exact structure and that the lattice of smaller exact structures of an admissible exact structure is Boolean.
