论文标题
与柔和的代数和群集代数的应用，交叉矢量与瓷砖上方
Intersection vectors over tilings with applications to gentle algebras and cluster algebras
论文作者
论文摘要
事实证明，在温和条件下，其相交载体在铺平上的多种允许的弧线是唯一决定的。这将经典的结果概括为具有三角形的明显表面。我们将此结果应用于$τ$的$τ$ tisting the Algebras和snominator猜想的群集代数理论。就柔和的代数而言，事实证明，在柔和的代数$ a $上，不同的$τ$ rigid $ a $ modules具有不同的尺寸向量，并且只有$ a $ $ a $没有与完全关系的均匀循环。对于群集代数，已经为$ \ mathbb {a} \ mathbb {b} \ Mathbb {c} $的类型代数建立了分母的猜想。
It is proved that a multiset of permissible arcs over a tiling is uniquely determined by its intersection vector under a mild condition. This generalizes a classical result over marked surfaces with triangulations. We apply this result to study $τ$-tilting theory of gentle algebras and denominator conjecture in cluster algebras. In the case of gentle algebras, it is proved that different $τ$-rigid $A$-modules over a gentle algebra $A$ have different dimension vectors if and only if $A$ has no even oriented cycle with full relations. For cluster algebras, the denominator conjecture has been established for cluster algebras of type $\mathbb{A}\mathbb{B}\mathbb{C}$.
