We relate decategorifications of Ozsv\'ath-Szab\'o's new bordered theory for knot Floer homology to representations of $\mathcal{U}_q(\mathfrak{gl}(1|1))$. Specifically, we consider two subalgebras $\mathcal{C}_r(n,\mathcal{S})$ and $\mathcal{C}_l(n,\mathcal{S})$ of Ozsv\'ath- Szab\'o's algebra $\mathcal{B}(n,\mathcal{S})$, and identify their Grothendieck groups with tensor products of representations $V$ and $V^*$ of $\mathcal{U}_q(\mathfrak{gl}(1|1))$, where $V$ is the vector representation. We identify the decategorifications of Ozsv\'ath-Szab\'o's DA bimodules for elementary tangles with corresponding maps between representations. Finally, when the algebras are given multi-Alexander gradings, we demonstrate a relationship between the decategorification of Ozsv\'ath-Szab\'o's theory and Viro's quantum relative $\mathcal{A}^1$ of the Reshetikhin-Turaev functor based on $\mathcal{U}_q(\mathfrak{gl}(1|1))$.

中文翻译：

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论 Ozsváth 和 Szabó 的结 Floer 同源性有界理论的去范畴化

我们将 Ozsv\'ath-Szab\'o 的新边界理论的去范畴化与 $\mathcal{U}_q(\mathfrak{gl}(1|1))$ 的表示联系起来。具体来说，我们考虑 Ozsv\'ath-Szab\' 的两个子代数 $\mathcal{C}_r(n,\mathcal{S})$ 和 $\mathcal{C}_l(n,\mathcal{S})$ o 的代数 $\mathcal{B}(n,\mathcal{S})$，并用 $\mathcal{U}_q(\mathfrak{ gl}(1|1))$，其中 $V$ 是向量表示。我们确定了 Ozsv\'ath-Szab\'o 的 DA 双模的去分类，用于基本缠结，并在表示之间具有相应的映射。最后，当代数被赋予多重亚历山大等级时，我们证明了 Ozsv\'ath-Szab\'o 理论的去范畴化与 Viro' 之间的关系