论文标题
$ _ \ infty $ $ $变形的Khovanov弧弧代数和Stroppel的猜想
A$_\infty$ deformations of extended Khovanov arc algebras and Stroppel's Conjecture
论文作者
论文摘要
扩展的khovanov弧代数$ \ mathrm {k} _m^n $是分级的联想代数，自然而然地出现在各种情况下，从结和链接同源性，低维拓扑拓扑和拓扑量化量子学和拓扑量化理论到表示理论和符号阶段。 C. Stroppel在她的ICM 2010中指出，$ \ Mathrm {k} _m^n $在一定范围内消失的大型Hochschild共同体组，这意味着代数$ \ Mathrm k_m^n $承认没有非遗传的A $ _ \ formor for hormation n nof for niment Intge Intge Intins Intins是那些属于那样的人。 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。这些变形也可以看作是基于Mak和Smith的最新工作，与Hilbert Shemes相关的Fukaya-Seidel类别的$ _ \ infty $变形。
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.
