发表机构
Beijing Jiaotong University（北京交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究构造了有理Dyck楼梯关联代数与A型高阶Auslander代数幂等拐角的显式倾斜等价，证明了互素正整数的Chapoton-Ladkani-Rognerud猜想，拓展了区间倾斜机制并实现了楼梯导出范畴的几何实现。
AI 中文摘要
我们构造了每个有理Dyck楼梯的关联代数与A型高阶Auslander代数的典范幂等拐角之间的显式倾斜等价。在互素情形下，该拐角与Xing引入的代数$B_0$一致。所得的Dyck-拐角等价为已知等价链提供了缺失的环节，从而证明了互素正整数的Chapoton-Ladkani-Rognerud猜想。该Dyck-拐角等价本身不要求互素性假设，且与复制代数兼容。我们的主要工具是Chapoton-Ladkani-Rognerud区间倾斜机制的线性范畴扩展。相对定理适用于有限$\boldsymbol{k}$-线性范畴，要求总范畴及其纤维具有有限整体维数。与关联范畴情形不同，它允许任意有限维$\text{Hom}$空间、非零态射的零复合，且不要求对角自同态代数是半单的。倾斜对象由纤维可表函子的正合右Kan扩张构造。我们计算其反向索引自同态范畴（包括所有强制零复合），进而计算其反向自同态代数。对每个坐标迭代该构造，得到每个有限坐标楼梯的关联代数与A型高阶Auslander代数的幂等拐角之间的显式导出等价。我们进一步将所得的楼梯导出范畴实现为停止圆盘对称积的部分包裹Fukaya范畴中由乘积拉格朗日量生成的三角化子范畴，在互素Dyck情形下则实现为对称Brieskorn-Pham奇点的Fukaya-Seidel范畴。
英文摘要
We construct an explicit tilting equivalence between the incidence algebra of every rational Dyck staircase and a canonical idempotent corner of a higher Auslander algebra of type~$A$. In the coprime case, this corner identifies with the algebra $B_0$ introduced by Xing. The resulting Dyck-corner equivalence supplies the missing link in the previously known chain of equivalences and thereby proves the Chapoton-Ladkani-Rognerud conjecture for coprime positive integers. The Dyck-corner equivalence itself requires no coprimality hypothesis and is compatible with replicated algebras. Our main tool is a linear-categorical extension of the interval-tilting mechanism of Chapoton-Ladkani-Rognerud. The relative theorem applies to finite $\kk$-linear categories under finite-global-dimension assumptions on the total category and its fibers. In contrast with the incidence-category setting, it allows arbitrary finite-dimensional $\Hom$ spaces and zero composites of nonzero morphisms, and it does not require the diagonal endomorphism algebras to be semisimple. The tilting object is constructed from exact right Kan extensions of fiberwise representables. We compute its opposite indexed endomorphism category, including all forced-zero compositions, and hence its opposite endomorphism algebra. Iterating this construction one coordinate at a time yields an explicit derived equivalence between the incidence algebra of every finite coordinate staircase and an idempotent corner of a higher Auslander algebra of type~$A$. We further realize the resulting staircase derived categories as triangulated subcategories generated by product Lagrangians in partially wrapped Fukaya categories of stopped-disk symmetric products and, in the coprime Dyck case, as Fukaya-Seidel categories of symmetric Brieskorn--Pham singularities.
CommentsThe title is changed, some results are modified to prove the main results. All comments are welcome!
