6.1.4. Gluing
The 2 2 Υ + \Upsilon^{+} Υ − \Upsilon^{-}
We consider the bimodule 2 2 E 1 = ⨁ s ≥ 0 L − ( s , 1 ) E_{1}=\bigoplus_{s\geq 0}L^{-}(s,1) E 2 = ⨁ s ≥ 0 L + ( s , 1 ) E_{2}=\bigoplus_{s\geq 0}L^{+}(s,1) E 1 ∨ → ∼ ⨁ s ≥ 0 R − ( s , 1 ) E_{1}^{\vee}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\bigoplus_{s\geq 0}R^{-}(s,1)
Define σ : E 2 E 1 → ∼ E 1 E 2 \sigma:E_{2}E_{1}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E_{1}E_{2} s ≥ 1 s\geq 1 ( H s − 1 , H s + 1 ) (H_{s-1},H_{s+1})
L + ( s − 1 , 1 ) ⊗ H s L − ( s , 1 ) \textstyle{L^{+}(s-1,1)\otimes_{H_{s}}L^{-}(s,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} σ \scriptstyle{\sigma} ∼ \scriptstyle{\sim} a ⊗ b ↦ a b \scriptstyle{a\otimes b\mapsto ab} ∼ \scriptstyle{\sim} L − ( s − 1 , 1 ) ⊗ H s L + ( s , 1 ) \textstyle{L^{-}(s-1,1)\otimes_{H_{s}}L^{+}(s,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces} a ⊗ b ↦ a b \scriptstyle{a\otimes b\mapsto ab} ∼ \scriptstyle{\sim} H s + 1 \textstyle{H_{s+1}}
Note that the left action of a ∈ H s − 1 a\in H_{s-1} H s + 1 H_{s+1} f 1 ( a ) f_{1}(a) 4.3.1
As in (5.3.1 σ \sigma
λ : R − ( − 1 , − , e ) ⊗ L + ( − , − 2 , e ) → L + ( − 1 , − , e ) ⊗ R − ( − , − 2 , e ) \lambda:R^{-}(-_{1},-,e)\otimes L^{+}(-,-_{2},e)\to L^{+}(-_{1},-,e)\otimes R^{-}(-,-_{2},e)
where
λ ( e s , e s ) \lambda(e^{s},e^{s}) ( H s , H s ) (H_{s},H_{s})
R − ( e s , − , e ) ⊗ L + ( − , e s , e ) = R − ( s − 1 , 1 ) ⊗ H s − 1 L + ( s − 1 , 1 ) \textstyle{R^{-}(e^{s},-,e)\otimes L^{+}(-,e^{s},e)=R^{-}(s-1,1)\otimes_{H_{s-1}}L^{+}(s-1,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces} a ⊗ b \textstyle{a\otimes b\ignorespaces\ignorespaces\ignorespaces\ignorespaces} L + ( e s , − , e ) ⊗ R − ( − , e s , e ) = L + ( s , 1 ) ⊗ H s + 1 R − ( s , 1 ) \textstyle{L^{+}(e^{s},-,e)\otimes R^{-}(-,e^{s},e)=L^{+}(s,1)\otimes_{H_{s+1}}R^{-}(s,1)} a ⊗ f 1 ( b ) \textstyle{a\otimes f_{1}(b)}
The morphism
( R − ( − , − , e ) λ L + ( − , − , e ) ) ∘ ( R − ( − , − , e ) 2 τ − τ L + ( − , − , e ) 2 ) : R − ( − , − , e ) 2 L + ( − , − , e ) 2 → ( R − ( − , − , e ) L + ( − , − , e ) ) 2 ⟨ − 1 ⟩ (R^{-}(-,-,e)\lambda L^{+}(-,-,e))\circ(R^{-}(-,-,e)^{2}\tau-\tau L^{+}(-,-,e)^{2}):\\
R^{-}(-,-,e)^{2}L^{+}(-,-,e)^{2}\to(R^{-}(-,-,e)L^{+}(-,-,e))^{2}\langle-1\rangle
is on ( e s , e s ) (e^{s},e^{s}) ( H s , H s ) (H_{s},H_{s})
R − ( s − 1 , 1 ) ⊗ H s − 1 R − ( s − 2 , 1 ) ⊗ H s − 2 L + ( s − 2 , 1 ) ⊗ H s − 1 L + ( s − 1 , 1 ) → R − ( s − 1 , 1 ) ⊗ H s − 1 L + ( s − 1 , 1 ) ⊗ H s R − ( s − 1 , 1 ) ⊗ H s − 1 L + ( s − 1 , 1 ) ⟨ − 1 ⟩ R^{-}(s-1,1)\otimes_{H_{s-1}}R^{-}(s-2,1)\otimes_{H_{s-2}}L^{+}(s-2,1)\otimes_{H_{s-1}}L^{+}(s-1,1)\to\\
R^{-}(s-1,1)\otimes_{H_{s-1}}L^{+}(s-1,1)\otimes_{H_{s}}R^{-}(s-1,1)\otimes_{H_{s-1}}L^{+}(s-1,1)\langle-1\rangle
given by
1 ⊗ 1 ⊗ 1 ⊗ 1 ↦ T 1 ⊗ 1 ⊗ 1 ⊗ 1 − 1 ⊗ 1 ⊗ 1 ⊗ T s − 1 . 1\otimes 1\otimes 1\otimes 1\mapsto T_{1}\otimes 1\otimes 1\otimes 1-1\otimes 1\otimes 1\otimes T_{s-1}.
Given s ≥ 1 s\geq 1 M s = R − ( s − 1 , 1 ) ⊗ H s − 1 L + ( s − 1 , 1 ) M_{s}=R^{-}(s-1,1)\otimes_{H_{s-1}}L^{+}(s-1,1) ( H s , H s ) (H_{s},H_{s}) s ≥ 2 s\geq 2 κ = 1 ⊗ 1 ⊗ 1 ⊗ T s − 1 − T 1 ⊗ 1 ⊗ 1 ⊗ 1 ∈ M s ⊗ H s M s \kappa=1\otimes 1\otimes 1\otimes T_{s-1}-T_{1}\otimes 1\otimes 1\otimes 1\in M_{s}\otimes_{H_{s}}M_{s} κ = 0 \kappa=0 s = 1 s=1 M 0 = 0 M_{0}=0
0P7C
Lemma 6.1.5 . There is a morphism of differential graded ( H s , H s ) (H_{s},H_{s}) M s → H ^ s + M_{s}\to\hat{H}_{s}^{+} a ⊗ b ↦ a c b a\otimes b\mapsto acb a , b ∈ H s a,b\in H_{s} ( H s , H s ) (H_{s},H_{s}) T H s ( M s ) / ( κ ) → ∼ H ^ s + T_{H_{s}}(M_{s})/(\kappa)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\hat{H}_{s}^{+}
0P7D
Proof. We have a c T i b = a T i + 1 c b acT_{i}b=aT_{i+1}cb i ∈ { 1 , … , s − 2 } i\in\{1,\ldots,s-2\}
We have now a morphism of differential graded algebras and of ( H s , H s ) (H_{s},H_{s}) f ′ : T H s ( M s ) → H ^ s + f^{\prime}:T_{H_{s}}(M_{s})\to\hat{H}_{s}^{+} M s → H ^ s + M_{s}\to\hat{H}_{s}^{+}
f ′ ( ( 1 ⊗ 1 ) ⊗ ( 1 ⊗ T s − 1 ) ) = c 2 T s − 1 = T 1 c 2 = f ′ ( ( T 1 ⊗ 1 ) ⊗ ( 1 ⊗ 1 ) ) , f^{\prime}((1\otimes 1)\otimes(1\otimes T_{s-1}))=c^{2}T_{s-1}=T_{1}c^{2}=f^{\prime}((T_{1}\otimes 1)\otimes(1\otimes 1)),
hence
f ′ ( κ ) = 0 f^{\prime}(\kappa)=0 f f f : T H s ( M s ) / ( κ ) → H ^ s + f:T_{H_{s}}(M_{s})/(\kappa)\to\hat{H}_{s}^{+}
On the other hand, H ^ s + \hat{H}_{s}^{+} H s H_{s} c c c T i = T i + 1 c cT_{i}=T_{i+1}c i ∈ { 1 , … , s − 2 } i\in\{1,\ldots,s-2\} c 2 T s − 1 = T 1 c 2 c^{2}T_{s-1}=T_{1}c^{2} 3.2.9 T i + 1 ⊗ 1 = 1 ⊗ T i T_{i+1}\otimes 1=1\otimes T_{i} M s M_{s} i ∈ { 1 , … , s − 2 } i\in\{1,\ldots,s-2\} ( 1 ⊗ 1 ) ⊗ ( 1 ⊗ T s − 1 ) = ( T 1 ⊗ 1 ) ⊗ ( 1 ⊗ 1 ) (1\otimes 1)\otimes(1\otimes T_{s-1})=(T_{1}\otimes 1)\otimes(1\otimes 1) M s ⊗ M s M_{s}\otimes M_{s} g : H ^ s + → T H s ( M s ) / ( κ ) , T i ↦ T i , c ↦ 1 ⊗ 1 g:\hat{H}_{s}^{+}\to T_{H_{s}}(M_{s})/(\kappa),\ T_{i}\mapsto T_{i},\ c\mapsto 1\otimes 1 f f g g
Let ℋ + {\mathcal{H}}^{+} 𝐙 ≥ 0 {\mathbf{Z}}_{\geq 0} Hom ℋ + ( m , n ) = δ m n 𝔖 ^ n + , nil \operatorname{Hom}\nolimits_{{\mathcal{H}}^{+}}(m,n)=\delta_{mn}\hat{{\mathfrak{S}}}_{n}^{+,\mathrm{nil}} 6.1.5
0P7E
Theorem 6.1.6 . The construction of Lemma 6.1.5 Θ : Δ λ ′ ( 𝒰 ∙ ) → ∼ ℋ + \Theta:\Delta^{\prime}_{\lambda}({\mathcal{U}}^{\bullet})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{H}}^{+}
Since σ \sigma 2 2 Δ λ ′ ( 𝒰 ∙ ) \Delta^{\prime}_{\lambda}({\mathcal{U}}^{\bullet}) 5.3.3 6.1.6 2 2 ℋ + {\mathcal{H}}^{+} H ^ n + \hat{H}_{n}^{+}
E n = H ^ n + [ 1 ] ⊕ H ^ n + h ↦ c h E_{n}={\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 22.47226pt\hbox{{\hbox{\kern-22.47226pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.73114pt\hbox{$\textstyle{\hat{H}_{n}^{+}[1]\oplus\hat{H}_{n}^{+}}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-9.46863pt\raise 22.8116pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.43056pt\hbox{$\scriptstyle{h\mapsto ch}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 14.2263pt\raise 11.38104pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}}
We define a left action of H ^ n − 1 + \hat{H}_{n-1}^{+} E n E_{n}
T i acts by ( T i 0 0 T i + 1 ) for 1 ≤ i ≤ n − 2 and c acts by ( c T n − 1 1 0 T 1 c ) . T_{i}\text{ acts by }\left(\begin{matrix}T_{i}&0\\
0&T_{i+1}\end{matrix}\right)\text{ for }1\leq i\leq n-2\text{ and }c\text{ acts by }\left(\begin{matrix}cT_{n-1}&1\\
0&T_{1}c\end{matrix}\right).
This defines a structure of differential graded
( H ^ n − 1 + , H ^ n + ) (\hat{H}_{n-1}^{+},\hat{H}_{n}^{+}) E n E_{n}
Note that setting E = 0 E=0 c c ℋ + {\mathcal{H}}^{+} Hom ℋ ( m , n ) = δ m n 𝔖 ^ n nil \operatorname{Hom}\nolimits_{{\mathcal{H}}}(m,n)=\delta_{mn}\hat{{\mathfrak{S}}}_{n}^{\mathrm{nil}}
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