Subsection 7.3.6. (0PHF) · ScalingStacks

Consider a chord diagram $({\mathcal{Z}},{\mathbf{a}})$ as in §7.2.4 with associated singular curve $Z$ . Define

{\mathcal{A}}(Z,1)=\operatorname{End}\nolimits_{\operatorname{add}\nolimits({\mathcal{S}}(Z,1))}(\bigoplus_{\begin{subarray}{c}z\in Z_{exc}\end{subarray}}z).

Proposition 7.3.19 shows that the algebra ${\mathcal{A}}(Z,1)$ is the opposite of Zarev’s algebra ${\mathcal{A}}_{Za}({\mathcal{Z}},1)$ [Za, Definition 2.6] (this will be explained for the more general algebras ${\mathcal{A}}(Z)$ in §7.4.11).

$\bullet\$ Consider the chord diagram $({\mathbf{R}},\{\{1,3\},\{2,4\}\})$ .

The associated singular curve $Z$ is the quotient of oriented ${\mathbf{R}}$ by the relation whose non-trivial equivalence classes are $1=\{1,3\}$ and $2=\{2,4\}$ .

The full pointed subcategory of ${\mathcal{S}}(Z,1)$ with object set $\{1,2\}$ is generated by $\alpha,\alpha^{\prime}:1\to 2$ and $\beta:2\to 1$ with relations $\beta\alpha=\alpha^{\prime}\beta=0$ . This corresponds to the well-known “torus algebra” in bordered Floer homology.

$\bullet\$ Consider the chord diagram $(S^{1},\{\{\pm 1\},\{\pm i\}\})$ .

The associated singular curve $Z$ is the quotient of oriented $S^{1}$ by the relation whose non trivial equivalence classes are $1=\{\pm 1\}$ and $2=\{\pm i\}$ .

The full pointed subcategory of ${\mathcal{S}}(Z,1)$ with object set $\{1,2\}$ is generated by $\alpha,\alpha^{\prime}:1\to 2$ and $\beta,\beta^{\prime}:2\to 1$ with relations $\beta\alpha=\alpha^{\prime}\beta=\alpha\beta^{\prime}=\beta^{\prime}\alpha^{\prime}=0$ . A curved $A_{\infty}$ -deformation of this subcategory appears in [LiOzTh3].

We have

\operatorname{End}\nolimits_{{\mathcal{S}}(Z,1)}(1)=\{\operatorname{id}\nolimits\}\sqcup\{(\beta^{\prime}\alpha\beta\alpha^{\prime})^{n}\}_{n\geq 1}\sqcup\{(\beta^{\prime}\alpha\beta\alpha^{\prime})^{n}\beta^{\prime}\alpha\}_{n\geq 0}\sqcup\{\beta\alpha^{\prime}(\beta^{\prime}\alpha\beta\alpha^{\prime})^{n})\}_{n\geq 0}\sqcup\{(\beta\alpha^{\prime}\beta^{\prime}\alpha)^{n})\}_{n\geq 1}

\operatorname{End}\nolimits_{{\mathcal{S}}(Z,1)}(2)=\{\operatorname{id}\nolimits\}\sqcup\{(\alpha^{\prime}\beta^{\prime}\alpha\beta)^{n}\}_{n\geq 1}\sqcup\{(\alpha^{\prime}\beta^{\prime}\alpha\beta)^{n}\alpha^{\prime}\beta^{\prime}\}_{n\geq 0}\sqcup\{\alpha\beta(\alpha^{\prime}\beta^{\prime}\alpha\beta)^{n})\}_{n\geq 0}\sqcup\{(\alpha\beta\alpha^{\prime}\beta^{\prime})^{n})\}_{n\geq 1}

\operatorname{Hom}\nolimits_{{\mathcal{S}}(Z,1)}(1,2)=\{\alpha^{\prime}(\beta^{\prime}\alpha\beta\alpha^{\prime})^{n}\}_{n\geq 0}\sqcup\{\alpha\beta\alpha^{\prime}(\beta^{\prime}\alpha\beta\alpha^{\prime})^{n}\}_{n\geq 0}\sqcup\{\alpha^{\prime}\beta^{\prime}\alpha(\beta\alpha^{\prime}\beta^{\prime}\alpha)^{n})\}_{n\geq 0}\sqcup\{\alpha(\beta\alpha^{\prime}\beta^{\prime}\alpha)^{n})\}_{n\geq 0}

\operatorname{Hom}\nolimits_{{\mathcal{S}}(Z,1)}(2,1)=\{(\beta^{\prime}\alpha\beta\alpha^{\prime})^{n}\beta^{\prime}\}_{n\geq 0}\sqcup\{(\beta^{\prime}\alpha\beta\alpha^{\prime})^{n}\beta^{\prime}\alpha\beta^{\prime}\}_{n\geq 0}\sqcup\{(\beta\alpha^{\prime}\beta^{\prime}\alpha)^{n})\beta\}_{n\geq 0}\sqcup\{(\beta\alpha^{\prime}\beta^{\prime}\alpha)^{n})\beta\alpha^{\prime}\beta^{\prime}\}_{n\geq 0}

7.3.6. One strand bordered algebras