论文标题
柔和和特殊的双词代数的真实Grothendieck组的非刚性区域
Non-rigid regions of real Grothendieck groups of gentle and special biserial algebras
论文作者
论文摘要
在一个领域的有限维代数$ a $的表示理论中，2-期（前）淤积络合物的分类是一个重要的问题。有用的工具之一是与真正的Grothendieck组中的2-期间派发复合物相关的G矢量锥体$ k_0（\ operatatorName {\ MathSf {proj}} a）_ {\ m athbb {r Mathbb {r}}} \ otimes _ {\ Mathbb {z}}} {\ Mathbb {r}} $。本文的目的是研究Union $ \ operatoTorname的补充$ \ perionatorName {\ Mathsf {nr}} $，所有G-vector锥的$ \ Mathsf {cone}} $，我们称之为非枪击区域。根据Iyama和我们的工作，$ \ peratatorName {\ Mathsf {nr}} $由2-期间的预选复合物和一定的封闭子集$ r_0 \ subset k_0（\ operatatorName {\ operatoTorname {\ m athsf {proj}} a）在本文中，我们对$ r_0 $的明确说明，用于完整的特殊双层代数，以$ a $的Gabriel Quiver中有限的最大非零路径。我们还证明，$ \ operatorName {\ mathsf {nr}} $具有某种分形属性，并且$ \ operatatorName {\ mathsf {nr}} $包含在数量的许多编辑中的超级计划中。因此，任何完整的特殊双性代数都是g-tame，也就是说，$ \ operatotorname {\ atmathsf {cone}} $在$ k_0（\ operatorname {\ propatatorname {\ propateAname {\ mathsf {proj}}} a）_ {\ mathbb {r {r}} $。
In the representation theory of finite-dimensional algebras $A$ over a field, the classification of 2-term (pre)silting complexes is an important problem. One of the useful tool is the g-vector cones associated to the 2-term presilting complexes in the real Grothendieck group $K_0(\operatorname{\mathsf{proj}} A)_{\mathbb{R}}:=K_0(\operatorname{\mathsf{proj}} A) \otimes_{\mathbb{Z}} {\mathbb{R}}$. The aim of this paper is to study the complement $\operatorname{\mathsf{NR}}$ of the union $\operatorname{\mathsf{Cone}}$ of all g-vector cones, which we call the non-rigid region. By the work of Iyama and us, $\operatorname{\mathsf{NR}}$ is determined by 2-term presilting complexes and a certain closed subset $R_0 \subset K_0(\operatorname{\mathsf{proj}} A)_{\mathbb{R}}$, which is called the purely non-rigid region. In this paper, we give an explicit description of $R_0$ for complete special biserial algebras in terms of a finite set of maximal nonzero paths in the Gabriel quiver of $A$. We also prove that $\operatorname{\mathsf{NR}}$ has some kind of fractal property and that $\operatorname{\mathsf{NR}}$ is contained in a union of countably many hyperplanes of codimension one. Thus, any complete special biserial algebra is g-tame, that is, $\operatorname{\mathsf{Cone}}$ is dense in $K_0(\operatorname{\mathsf{proj}} A)_{\mathbb{R}}$.
