A$_\infty$变形的扩展Khovanov弧代数与Stroppel猜想
A$_\infty$ deformations of extended Khovanov arc algebras and Stroppel's conjecture
AI总结：
本文证明扩展Khovanov弧代数$\mathrm{K}_m^n$存在非平凡的A$_\infty$变形，反驳了Stroppel的猜想，并提供了显式的代数构造方法。
摘要：
扩展的Khovanov弧代数$\mathrm{K}_m^n$是带权的结合代数，自然出现在结和链环同调、低维拓扑、拓扑量子场论、表示论和辛几何等多种背景下。C. Stroppel在2010年ICM演讲中猜想，$\mathrm{K}_m^n$的大梯度Hochschild上同调群在特定范围内消失，这意味着$\mathrm{K}_m^n$代数没有非平凡的A$_\infty$变形，尤其表明这些代数本质上是形式的。尽管Stroppel的猜想已由Seidel和Thomas的工作证实对$\mathrm{K}_m^1$和$\mathrm{K}_1^n$成立，我们证明对于所有$m, n \geq 2$，$\mathrm{K}_m^n$确实存在非平凡的A$_\infty$变形，其更高乘积不为零。我们具体描述了$\mathrm{K}_m^n$及其Koszul对偶作为带有关系的有向图路径代数，并通过利用A$_\infty$代数与Koszul代数的Koszul对偶的滤过结合代数变形之间的对应关系，给出了$\mathrm{K}_m^n$的A$_\infty$变形的显式代数构造。这些变形也可以视为Fukaya-Seidel范畴的A$_\infty$变形，这些范畴基于Mak和Smith最近的工作。
英文摘要：
Extended Khovanov arc algebras $\mathrm{K}_m^n$ are graded associative algebras which naturally appear in a variety of contexts, from knot and link homology, low-dimensional topology and topological quantum field theory to representation theory and symplectic geometry. C. Stroppel conjectured in her ICM 2010 address that the bigraded Hochschild cohomology groups of $\mathrm{K}_m^n$ vanish in a certain range, implying that the algebras $\mathrm K_m^n$ admit no nontrivial A$_\infty$ deformations, in particular that the algebras are intrinsically formal. Whereas Stroppel's Conjecture is known to hold for the algebras $\mathrm K_m^1$ and $\mathrm K_1^n$ by work of Seidel and Thomas, we show that $\mathrm K_m^n$ does in fact admit nontrivial A$_\infty$ deformations with nonvanishing higher products for all $m, n \geq 2$. We describe both $\mathrm K_m^n$ and its Koszul dual concretely as path algebras of quivers with relations and give an explicit algebraic construction of A$_\infty$ deformations of $\mathrm K_m^n$ by using the correspondence between A$_\infty$ deformations of a Koszul algebra and filtered associative deformations of its Koszul dual. These deformations can also be viewed as A$_\infty$ deformations of Fukaya--Seidel categories associated to Hilbert schemes of surfaces based on recent work of Mak and Smith.
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