7.4.11. Bordered Heegaard Floer algebras
We consider a chord diagram ( 𝒵 , 𝐚 ) ({\mathcal{Z}},{\mathbf{a}}) 7.2.4 Z 1 , … , Z l Z_{1},\ldots,Z_{l} 𝒵 {\mathcal{Z}} 𝐚 ~ = ⋃ { z , z ′ } ∈ 𝐚 { z , z ′ } \tilde{{\mathbf{a}}}=\bigcup_{\{z,z^{\prime}\}\in{\mathbf{a}}}\{z,z^{\prime}\} n i = | 𝐚 ~ ∩ Z i | n_{i}=|\tilde{{\mathbf{a}}}\cap Z_{i}| q : Z ~ → Z q:\tilde{Z}\to Z
The isomorphism (7.4.5 Z ~ = Z ̊ 1 ∐ ⋯ ∐ Z ̊ l \tilde{Z}=\mathring{Z}_{1}\coprod\cdots\coprod\mathring{Z}_{l} 6.3.2 7.4.33
𝒜 ( n 1 ) ⊗ ⋯ ⊗ 𝒜 ( n l ) → ∼ End add ( 𝒮 ( Z ~ ) ) ( ⨁ I ⊂ 𝐚 ~ I ) . {\mathcal{A}}(n_{1})\otimes\cdots\otimes{\mathcal{A}}(n_{l})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{End}\nolimits_{\operatorname{add}\nolimits({\mathcal{S}}(\tilde{Z}))}(\bigoplus_{I\subset\tilde{{\mathbf{a}}}}I).
It is compatible with the gradings, via the embedding
G ′ ( n 1 ) × ⋯ × G ′ ( n l ) ↪ Γ 𝐚 ( Z ~ ) G^{\prime}(n_{1})\times\cdots\times G^{\prime}(n_{l})\hookrightarrow\Gamma_{{\mathbf{a}}}(\tilde{Z}) 6.3.2 7.4.8
The differential algebra 𝒜 ( 𝒵 ) {\mathcal{A}}({\mathcal{Z}}) 𝒵 {\mathcal{Z}} ( G ′ ( n 1 ) × ⋯ × G ′ ( n l ) ) (G^{\prime}(n_{1})\times\cdots\times G^{\prime}(n_{l})) 𝒜 ( n 1 ) ⊗ ⋯ ⊗ 𝒜 ( n l ) {\mathcal{A}}(n_{1})\otimes\cdots\otimes{\mathcal{A}}(n_{l}) [Za , Definition 2.6] and [LiOzTh1 , Definition 3.23] for the original
setting where l = 1 l=1
𝒜 ( 𝒵 ) → ∼ End add ( 𝒮 ( Z ) ) ( ⨁ S ⊂ 𝐚 S ) {\mathcal{A}}({\mathcal{Z}})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{End}\nolimits_{\operatorname{add}\nolimits({\mathcal{S}}(Z))}\bigl(\bigoplus_{S\subset{\mathbf{a}}}S\bigr)
making the following diagram commutative
𝒜 ( 𝒵 ) \textstyle{{\mathcal{A}}({\mathcal{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} ∼ \scriptstyle{\sim} End add ( 𝒮 ( Z ) ) ( ⨁ S ⊂ 𝐚 S ) \textstyle{\operatorname{End}\nolimits_{\operatorname{add}\nolimits({\mathcal{S}}(Z))}\bigl(\bigoplus_{S\subset{\mathbf{a}}}S\bigr)\ignorespaces\ignorespaces\ignorespaces\ignorespaces} q # \scriptstyle{q^{\#}} 𝒜 ( n 1 ) ⊗ ⋯ ⊗ 𝒜 ( n l ) \textstyle{{\mathcal{A}}(n_{1})\otimes\cdots\otimes{\mathcal{A}}(n_{l})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} ∼ \scriptstyle{\sim} End add ( 𝒮 ( Z ~ ) ) ( ⨁ I ⊂ 𝐚 ~ I ) \textstyle{\operatorname{End}\nolimits_{\operatorname{add}\nolimits({\mathcal{S}}(\tilde{Z}))}(\bigoplus_{I\subset\tilde{{\mathbf{a}}}}I)}