We show that Ozsvath-Szabo's bordered algebra used to efficiently compute knot Floer homology is a graded flat deformation of the regular block of a $\mathfrak{q}$-presentable quotient of parabolic category $\mathcal{O}$. We identify the endomorphism algebra of a minimal projective generator for this block with an explicit quotient of the Ozsvath-Szabo algebra using Sartori's diagrammatic formulation of the endomorphism algebra. Both of these algebras give rise to categorifications of tensor products of the vector representation $V^{\otimes n}$ for $U_q(\mathfrak{gl}(1|1))$. Our isomorphism allows us to transport a number of constructions between these two algebras, leading to a new (fully) diagrammatic reinterpretation of Sartori's algebra, new modules over Ozsvath-Szabo's algebra lifting various bases of $V^{\otimes n}$, and bimodules over Ozsvath-Szabo's algebra categorifying the action of the quantum group element $F$ and its dual on $V^{\otimes n}$.

中文翻译：

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Ozsváth-Szabó 边界代数和 O 类的子商

我们表明，用于有效计算结 Floer 同源性的 Ozsvath-Szabo 的边界代数是抛物线范畴 $\mathcal{O}$ 的 $\mathfrak{q}$ 可表示商的规则块的分级平面变形。我们使用 Sartori 的自同态代数图解公式，使用 Ozsvath-Szabo 代数的显式商来确定该块的最小射影生成器的自同态代数。这两个代数都对 $U_q(\mathfrak{gl}(1|1))$ 的向量表示 $V^{\otimes n}$ 的张量积进行分类。我们的同构允许我们在这两个代数之间传输许多构造，导致对 Sartori 代数的新（完全）图解重新解释，在 Ozsvath-Szabo 代数上的新模块提升了 $V^{\otimes n}$ 的各种基础，