The circle quantum group and the infinite root stack of a curve

Once you want to free your mind about a concept of harmony and music being correct, you can do whatever you want.

Giorgio Moroder

Abstract

In the present paper, we give a definition of the quantum group $\mathbf {U}_\upsilon (\mathfrak {sl}(S^1))$ of the circle $S^1:={\mathbb {R}}/{\mathbb {Z}}$, and its fundamental representation. Such a definition is motivated by a realization of a quantum group $\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$ associated to the rational circle $S^1_{\mathbb {Q}}:={\mathbb {Q}}/{\mathbb {Z}}$ as a direct limit of $\mathbf {U}_\upsilon (\widehat{\mathfrak {sl}}(n))$’s, where the order is given by divisibility of positive integers. The quantum group $\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$ arises as a subalgebra of the Hall algebra of coherent sheaves on the infinite root stack $X_\infty$ over a fixed smooth projective curve X defined over a finite field. Via this Hall algebra approach, we are able to realize geometrically the fundamental and the tensor representations, and a family of symmetric tensor representations, depending on the genus $g_X$, of $\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$. Moreover, we show that $\mathbf {U}_\upsilon (\widehat{\mathfrak {sl}}(+\infty ))$ and $\mathbf {U}_\upsilon (\widehat{\mathfrak {sl}}(\infty ))$ are subalgebras of $\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$. As proved by T. Kuwagaki in the appendix, the quantum group $\mathbf {U}_\upsilon (\mathfrak {sl}(S^1))$ naturally arises as well in the mirror dual picture, as a Hall algebra of constructible sheaves on the circle $S^1$.

Appendices

Appendix A: Some results from [36]

In this section we provide the proofs of some of the results in [36], which are used in the main body of the paper. This is done in order to avoid possible confusions since our assumptions differ sometimes from loc.cit. (for example, the orientation of the cyclic quiver we chose in the main body of the paper is the opposite of the one in loc.cit.). We will follow the notation introduced in Sects. 3.1 and 4.

Lemma A.1

Let $1\le i\le n$ and $1<j<n$. Then

$$\begin{aligned} 1_{{}_n \mathcal {S}_i^{(j)}}=\upsilon \, 1_{{}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}}\, 1_{{}_n \mathcal {S}_{i}^{(j-1)}} -1_{{}_n \mathcal {S}_{i}^{(j-1)}}\, 1_{{}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}}. \end{aligned}$$

Here, we set formally ${}_n\mathcal {S}_0^{(\ell )}:={}_n \mathcal {S}_n^{(\ell )}$ for any positive integer $\ell$.

Proof

First note, that because of our assumption on j, we have

$$\begin{aligned} \langle [{}_n \mathcal {S}_i^{(j-1)}], [{}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}]\rangle&= 0\\ \langle [{}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}], [{}_n \mathcal {S}_i^{(j-1)}]\rangle&= {\left\{ \begin{array}{ll} -1 &{} \text {if }\,\,{}_n \{ i+1-j \}=0,\\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

By applying the functor $\mathsf {Hom}(\cdot , {}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)})$ to the short exact sequence

$$\begin{aligned} 0\rightarrow \mathcal {L}_n^{\otimes \, -i}\rightarrow \mathcal {L}_n^{\otimes \, j-1-i}\rightarrow {}_n \mathcal {S}_i^{(j-1)}\rightarrow 0, \end{aligned}$$

(A.1)

we obtain

$$\begin{aligned} 0\rightarrow \mathsf {Hom}({}_n \mathcal {S}_i^{(j-1)}, {}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)})\rightarrow \mathsf {Hom}(\mathcal {L}_n^{\otimes \, j-1-i}, {}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)})\rightarrow \cdots \end{aligned}$$

Since $\mathsf {Hom}(\mathcal {L}_n^{\otimes \, j-1-i}, {}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)})\simeq H^0(X_n, {}_n\mathcal {S}_n^{(1)})=0$, we get

$$\begin{aligned} \mathsf {Hom}({}_n \mathcal {S}_i^{(j-1)}, {}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)})=0 , \ \mathsf {Ext}^1({}_n \mathcal {S}_i^{(j-1)}, {}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)})=0. \end{aligned}$$

Now, by applying the functor $\mathsf {Hom}({}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}, \cdot )$ to the short exact sequence (A.1), we obtain

$$\begin{aligned} \cdots \rightarrow \mathsf {Ext}^1({}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}, \mathcal {L}_n^{\otimes \, j-1-i})\rightarrow \mathsf {Ext}^1({}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}, {}_n \mathcal {S}_i^{(j-1)})\rightarrow 0 . \end{aligned}$$

In this case, by Serre duality $\mathsf {Ext}^1({}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}, \mathcal {L}_n^{\otimes \, j-1-i})\simeq H^0(X_n, {}_n \mathcal {S}_1^{(1)})^\vee \simeq k$, we get

$$\begin{aligned} \mathsf {Hom}({}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}, {}_n \mathcal {S}_i^{(j-1)})=0 , \ \mathsf {Ext}^1({}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}, {}_n \mathcal {S}_i^{(j-1)})\simeq k. \end{aligned}$$

Thus we get

$$\begin{aligned} 1_{{}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}}\, 1_{{}_n \mathcal {S}_{i}^{(j-1)}}&=\upsilon ^{-1}\Big (1_{{}_n \mathcal {S}_i^{(j)}}+1_{{}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}\oplus {}_n \mathcal {S}_{i}^{(j-1)}}\Big ),\\ 1_{{}_n \mathcal {S}_{i}^{(j-1)}}\, 1_{{}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}}&=1_{{}_n \mathcal {S}_{{}_n \{ i+1-j \}}^{(1)}\oplus {}_n \mathcal {S}_{i}^{(j-1)}} . \end{aligned}$$

$\square$

By using similar arguments as in the proof above, one can also prove the following lemma.

Lemma A.2

Let $0\le j\le n-1$ and $m\in {\mathbb {Z}}_{>0}$. Then

$$\begin{aligned} 1_{{}_n \mathcal {S}_j^{(m n+j)}}= {\left\{ \begin{array}{ll} 1_{{}_n \mathcal {S}_n^{(m n)}} &{} \text {if }\,\,j=0, \\ 1_{{}_n \mathcal {S}_j^{(j)}}\, 1_{{}_n \mathcal {S}_j^{(m n)}}-\upsilon ^{-2}\,1_{{}_n \mathcal {S}_j^{(m n)}}\, 1_{{}_n \mathcal {S}_j^{(j)}} &{} \text {otherwise} . \end{array}\right. } \end{aligned}$$

Lemma A.3

Let $1\le i\le n$ and $d\in {\mathbb {Z}}$. Then

$$\begin{aligned} 1_{{}_n \mathcal {S}_i}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}=\upsilon ^{\delta _{n\{(i+d-1)/n\},0}-\delta _{n\{(i+d)/n\},0}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\, 1_{{}_n \mathcal {S}_{i}}+\delta _{n\{(i+d)/n\},0}\, \upsilon ^{-1}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d+1}. \end{aligned}$$

Proof

Let M be a line bundle on X and $1\le k\le n$ an integer. First compute $1_{{}_n \mathcal {S}_k}\, 1_{\pi _n^*M}$. Note that

$$\begin{aligned} \langle \overline{{}_n \mathcal {S}_k}, \overline{\pi _n^*M}\rangle&=\langle \overline{{}_n \mathcal {S}_k}, \overline{\mathcal {O}}_{X_n}+\deg (M)\, \delta _n\rangle =-\delta _{k,n},\\ \langle \overline{\pi _n^*M}, \overline{{}_n \mathcal {S}_k}\rangle&=\langle \overline{\mathcal {O}}_{X_n}+\deg (M)\, \delta _n, \overline{{}_n \mathcal {S}_k} \rangle =\delta _{k,1}. \end{aligned}$$

In addition,

$$\begin{aligned} \mathsf {Hom}({}_n \mathcal {S}_k, \pi _n^*M)&\simeq \mathsf {Ext}^1(\pi _n^*M, {}_n \mathcal {S}_k\otimes \omega _{X_n})^\vee =0\\ \mathsf {Ext}^1(\pi _n^*M, {}_n \mathcal {S}_k)&=0. \end{aligned}$$

Hence,

$$\begin{aligned} 1_{\pi _n^*M}\, 1_{{}_n \mathcal {S}_k}&=\upsilon ^{\delta _{k,1}}\, 1_{{}_n \mathcal {S}_k\oplus \pi _n^*M} ,\\ 1_{{}_n \mathcal {S}_k}\, 1_{\pi _n^*M}&= {\left\{ \begin{array}{ll} \upsilon ^{-1}\big (1_{{}_n \mathcal {S}_k\oplus \pi _n^*M}+1_{\pi _n^*M\otimes \mathcal {L}_n}\big ) &{} \text { for }\,\,k=n ,\\ \upsilon ^{\, 2\delta _{k,1}}\, 1_{{}_n \mathcal {S}_k\oplus \pi _n^*M} &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

Thus

$$\begin{aligned} 1_{{}_n \mathcal {S}_k}\, 1_{\pi _n^*M}&=\upsilon ^{\delta _{k,1}-\delta _{k,n}}\, 1_{\pi _n^*M}\, 1_{{}_n \mathcal {S}_k}+\delta _{k,n}\, \upsilon ^{-1}\, 1_{\pi _n^*M\otimes \mathcal {L}_n} \\&=\upsilon ^{\delta _{n\{k/n\},1}-\delta _{n\{k/n\},0}}\, 1_{\pi _n^*M}\, 1_{{}_n \mathcal {S}_k}+\delta _{n\{k/n\},0}\, \upsilon ^{-1}\, 1_{\pi _n^*M\otimes \mathcal {L}_n}. \end{aligned}$$

Now,

$$\begin{aligned}&1_{{}_n \mathcal {S}_i}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d} = 1_{{}_n \mathcal {S}_i}\, \sum _{\genfrac{}{}{0.0pt}{}{M\in \mathsf {Pic}(X)}{\deg (M)={}_n \lfloor d \rfloor }}\, 1_{\pi _n^*M\otimes \mathcal {L}_n^{\otimes \, {}_n \{ d \}}}=\sum _{\genfrac{}{}{0.0pt}{}{M\in \mathsf {Pic}(X)}{\deg (M)={}_n \lfloor d \rfloor }}\,1_{{}_n \mathcal {S}_i}\,1_{\pi _n^*M\otimes \mathcal {L}_n^{\otimes \, {}_n \{ d \}}}\\&\quad =\sum _{\genfrac{}{}{0.0pt}{}{M\in \mathsf {Pic}(X)}{\deg (M)={}_n \lfloor d \rfloor }}\, T_n^{{}_n \{ d \}}\big (1_{{}_n \mathcal {S}_{i+{}_n \{ d \}}}\,1_{\pi _n^*M}\big )\\&\quad =\sum _{\genfrac{}{}{0.0pt}{}{M\in \mathsf {Pic}(X)}{\deg (M)={}_n \lfloor d \rfloor }}\, T_n^{{}_n \{ d \}}\big (\upsilon ^{\delta _{{}_n \{ i+d \},1}-\delta _{{}_n \{ i+d \},0}}\, 1_{\pi _n^*M}\, 1_{{}_n \mathcal {S}_{i+{}_n \{ d \}}}+\delta _{{}_n \{ i+d \},0}\, \upsilon ^{-1}\, 1_{\pi _n^*M\otimes \mathcal {L}_n}\big ), \end{aligned}$$

and we get the assertion. $\square$

Thanks to the previous lemmas, we can provide a characterization of the commutation relations between the $1_{{}_n \mathcal {S}_i^{(j)}}$’s and the ${}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}$’s.

Proposition A.4

Let $1\le i \le n$ and $1\le j\le n-1$ be integers, let $d\in {\mathbb {Z}}$. Then

$$\begin{aligned}&1_{{}_n \mathcal {S}_i^{(j)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\\&\quad ={\left\{ \begin{array}{ll} \upsilon ^{-1}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\, 1_{{}_n \mathcal {S}_i^{(j)}}+\upsilon ^{-1}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d+j} &{} \text {if } \,\,{}_n \{ i+d \}=0, \\ {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\, 1_{{}_n \mathcal {S}_i^{(j)}} &{} \text {if } \,\,{}_n \{ i+d \}>j \text { and } {}_n \{ i+d \}\ne 0 ,\\ \upsilon \, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\, 1_{{}_n \mathcal {S}_i^{(j)}} &{} \text {if } \,\,{}_n \{ i+d \}=j \text { and } {}_n \{ i+d \}\ne 0, \\ {}_n \mathbf {1}^{\mathbf {ss}}_{1,\, d}\, 1_{{}_n \mathcal {S}_i^{(j)}}+(\upsilon -\upsilon ^{-1})\, {}_n \mathbf {1}^{\mathbf {ss}}_{1,\, {}_n \lfloor d \rfloor n+j-{}_n \{ i+d \}}\, 1_{{}_n \mathcal {S}_{{}_n \{ i+d \}}^{({}_n \{ i+d \})}} &{} \text {if }\,\, {}_n \{ i+d \}<j \text { and } {}_n \{ i+d \}\ne 0. \end{array}\right. } \end{aligned}$$

Proof

First, let us assume that $d=\alpha n$ for some $\alpha \in {\mathbb {Z}}$ and let $k\in \{1, \ldots , n\}$. Then the statement reduces to

$$\begin{aligned} 1_{{}_n \mathcal {S}_k^{(j)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}= {\left\{ \begin{array}{ll} \upsilon ^{-1}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\, 1_{{}_n \mathcal {S}_k^{(j)}}+\upsilon ^{-1}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n+j} &{} \text {if }\,\,k=n , \\ {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\, 1_{{}_n \mathcal {S}_k^{(j)}} &{} \text {if }\,\,n-1\ge k>j,\\ \upsilon \, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\, 1_{{}_n \mathcal {S}_k^{(j)}} &{} \text {if }\,\,k=j, \\ {}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\, 1_{{}_n \mathcal {S}_k^{(j)}}+(\upsilon -\upsilon ^{-1})\, {}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n+j-k}\, 1_{{}_n \mathcal {S}_k^{(k)}} &{}\text {if }\,\,k<j. \end{array}\right. } \end{aligned}$$

Assume that it is true for all $j'< j$ and let us prove it for j, by using Lemmas A.1 and A.3. For $n-1\ge k >j$, we have

$$\begin{aligned} 1_{{}_n \mathcal {S}_k^{(j)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}&=\Big (\upsilon \, 1_{{}_n \mathcal {S}_{{}_n \{ k+1-j \}}^{(1)}}\, 1_{{}_n \mathcal {S}_{k}^{(j-1)}} -1_{{}_n \mathcal {S}_{k}^{(j-1)}}\, 1_{{}_n \mathcal {S}_{{}_n \{ k+1-j \}}^{(1)}}\Big )\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\\&=\upsilon \, 1_{{}_n \mathcal {S}_{{}_n \{ k+1-j \}}^{(1)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\, 1_{{}_n \mathcal {S}_{k}^{(j-1)}}-1_{{}_n \mathcal {S}_{k}^{(j-1)}}\Big ({}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\, 1_{{}_n \mathcal {S}_{{}_n \{ k+1-j \}}^{(1)}}\Big )\\&={}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\, 1_{{}_n \mathcal {S}_k^{(j)}}. \end{aligned}$$

For $k=n$ and $j>1$, we get

$$\begin{aligned} 1_{{}_n \mathcal {S}_k^{(j)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}&=\Big (\upsilon \, 1_{{}_n \mathcal {S}_{n-j+1}^{(1)}}\, 1_{{}_n \mathcal {S}_{n}^{(j-1)}} -1_{{}_n \mathcal {S}_{n}^{(j-1)}}\, 1_{{}_n \mathcal {S}_{n-j+1}^{(1)}}\Big )\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\\&= \upsilon \, 1_{{}_n \mathcal {S}_{n-j+1}^{(1)}}\, \Big (\upsilon ^{-1}\, \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n} \, 1_{{}_n \mathcal {S}_n^{(j-1)}}+\upsilon ^{-1}\, \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n+j-1}\Big )-1_{{}_n \mathcal {S}_n^{(j-1)}}\, \Big ({}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\, 1_{{}_n \mathcal {S}_{n-j+1}^{(1)}}\Big )\\&=\upsilon ^{-1}\, {}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\, 1_{{}_n \mathcal {S}_n^{(j)}}+T_n^{j-1}\Big (1_{{}_n \mathcal {S}_{n}^{(1)}}\,{}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\Big )-\upsilon ^{-1}\, {}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n+j-1}\, 1_{{}_n \mathcal {S}_{n-j+1}^{(1)}}\\&=\upsilon ^{-1}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\, 1_{{}_n \mathcal {S}_k^{(j)}}+\upsilon ^{-1}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n+j} . \end{aligned}$$

For $j=k+1$, we obtain

$$\begin{aligned} 1_{{}_n \mathcal {S}_k^{(j)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}&=\Big (\upsilon \, 1_{{}_n \mathcal {S}_{n}^{(1)}}\, 1_{{}_n \mathcal {S}_{k}^{(k)}} -1_{{}_n \mathcal {S}_{k}^{(k)}}\, 1_{{}_n \mathcal {S}_{n}^{(1)}}\Big )\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\\&=\upsilon \, 1_{{}_n\mathcal {S}_n^{(1)}}\, \Big (\upsilon \, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n} \, 1_{{}_n \mathcal {S}_k^{(k)}}\Big )-1_{{}_n \mathcal {S}_k^{(k)}}\Big (\upsilon ^{-1}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n} \, 1_{{}_n \mathcal {S}_n^{(1)}}+\upsilon ^{-1}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n+1}\Big )\\&=\Big (\upsilon ^{-1}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n} \, 1_{{}_n \mathcal {S}_n^{(1)}}+\upsilon ^{-1}\,{}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n+1}\Big )\, 1_{{}_n \mathcal {S}_k^{(k)}}- {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n} \,1_{{}_n \mathcal {S}_k^{(k)}}\, 1_{{}_n \mathcal {S}_n^{(1)}}\\&\quad -\upsilon ^{-1}\, T_n\Big (1_{{}_n \mathcal {S}_{k+1}^{(k)}}\,{}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\Big )={}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\, 1_{{}_n \mathcal {S}_k^{(j)}}+(\upsilon -\upsilon ^{-1})\, {}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n+1}\, 1_{{}_n \mathcal {S}_k^{(k)}}, \end{aligned}$$

while for $k<j-1$, one has

$$\begin{aligned} 1_{{}_n \mathcal {S}_k^{(j)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}&= \Big (\upsilon \, 1_{{}_n \mathcal {S}_{n+k+1-j}^{(1)}}\, 1_{{}_n \mathcal {S}_{k}^{(j-1)}} -1_{{}_n \mathcal {S}_{k}^{(j-1)}}\, 1_{{}_n \mathcal {S}_{n+k+1-j}^{(1)}}\Big )\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\\&= \upsilon \, 1_{{}_n \mathcal {S}_{n+k+1-j}^{(1)}}\Big ({}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\, 1_{{}_n \mathcal {S}_k^{(j-1)}}+(\upsilon -\upsilon ^{-1})\, {}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n+j-1-k}\, 1_{{}_n \mathcal {S}_k^{(k)}}\Big )\\&\quad -1_{{}_n \mathcal {S}_k^{(j-1)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n} \, 1_{{}_n \mathcal {S}_{n+k+1-j}^{(1)}} \\&={}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\, 1_{{}_n \mathcal {S}_k^{(j)}}+(\upsilon -\upsilon ^{-1})\, {}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n+j-k}\, 1_{{}_n \mathcal {S}_k^{(k)}}\\&\quad +(\upsilon -\upsilon ^{-1})\, {}_n \mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n+j-1-k}\, \Big (1_{{}_n \mathcal {S}_{n+k+1-j}^{(1)}}\, 1_{{}_n \mathcal {S}_k^{(k)}}-1_{{}_n \mathcal {S}_k^{(k)}}\,1_{{}_n \mathcal {S}_{n+k+1-j}^{(1)}}\Big ) . \end{aligned}$$

Because of the assumptions on j and since $j-1>k$, we have

$$\begin{aligned} 1_{{}_n \mathcal {S}_{n+k+1-j}^{(1)}}\, 1_{{}_n \mathcal {S}_k^{(k)}}=1_{{}_n \mathcal {S}_k^{(k)}}\,1_{{}_n \mathcal {S}_{n+k+1-j}^{(1)}}, \end{aligned}$$

hence we get the result. The case $k=j$ is straightforward.

Finally, the assertion follows from the fact that

$$\begin{aligned} 1_{{}_n \mathcal {S}_k^{(j)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}=1_{{}_n \mathcal {S}_k^{(j)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n+\beta }=T_n^{\,\beta }\Big (1_{{}_n \mathcal {S}_{k+\beta }^{(j)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\Big ) \end{aligned}$$

for $d\in {\mathbb {Z}}$ such that $d=\alpha n+\beta$, with $\alpha , \beta \in {\mathbb {Z}}$ and $0\le \beta \le n-1$. $\square$

Corollary A.5

Let $1\le i \le n$ and $1\le j\le n-1$ be integers, let $d\in {\mathbb {Z}}$. Then

$$\begin{aligned} \omega _n\big (1_{{}_n \mathcal {S}_i^{(j)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\big )= {\left\{ \begin{array}{ll} \upsilon ^{-1}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d+j} &{} \text {if }\,\, {}_n \{ i+d \}=0, \\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

Lemma A.6

Let $1\le i\le n$ and $m\ne 0$ be integers and $d\in {\mathbb {Z}}$. Then

$$\begin{aligned} \omega _n\big (1_{{}_n \mathcal {S}_i^{(mn)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\big )= {\left\{ \begin{array}{ll} \upsilon ^{-m}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d+mn} &{} \text {if }\,\, {}_n \{ i+d \}=0, \\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

Proof

Let M be a line bundle on X. Let us first compute $1_{{}_n \mathcal {S}_i^{(mn)}}\, 1_{\pi _n^*M}$. By definition

$$\begin{aligned} 1_{{}_n \mathcal {S}_i^{(mn)}}\, 1_{\pi _n^*M}=\upsilon ^{\langle [{}_n \mathcal {S}_i^{(mn)}], [\pi _n^*M]\rangle }\, \sum _\mathcal {E}\, \frac{\mathbf {P}_{{}_n \mathcal {S}_i^{(m)}, \pi _n^*M}^\mathcal {E}}{\mathbf {a}({}_n \mathcal {S}_i^{(mn)}) \mathbf {a}(\pi _n^*M)}\, 1_\mathcal {E}, \end{aligned}$$

where we denote by $\mathbf {P}_{\mathcal {E}_1, \mathcal {E}_2}^\mathcal {F}$ the cardinality of the set of short exact sequences $0\rightarrow \mathcal {E}_2\rightarrow \mathcal {F}\rightarrow \mathcal {E}_1\rightarrow 0$ for any fixed triple of coherent sheaves $\mathcal {F}, \mathcal {E}_1, \mathcal {E}_2$; by $\mathbf {a}(\mathcal {E})$ the cardinality of the automorphism group of a coherent sheaf $\mathcal {E}$. Now, $\langle [{}_n \mathcal {S}_i^{(mn)}], [\pi _n^*M]\rangle =-m$,

$$\begin{aligned} \frac{\mathbf {P}_{{}_n \mathcal {S}_i^{(mn)}, \pi _n^*M}^\mathcal {E}}{\mathbf {a}(\pi _n^*M)}= {\left\{ \begin{array}{ll} \# \mathsf {Hom}^{\mathsf {surj}}(\mathcal {O}_{X_n}, {}_n\mathcal {S}_n^{(mn)})=q^m-q^{m-1} &{} \text {if }\,\,i=n,\\ 0 &{} \text {otherwise} , \end{array}\right. } \end{aligned}$$

and $\mathbf {a}({}_n \mathcal {S}_n^{(mn)})=q^m-q^{m-1}$. Thus, $\omega _n\big (1_{{}_n \mathcal {S}_i^{(mn)}}\, 1_{\pi _n^*M}\big )=\upsilon ^{-m}\, 1_{\pi _n^*M\otimes \mathcal {L}_n^{\otimes \, mn}}$. Therefore

$$\begin{aligned} \omega _n\big (1_{{}_n \mathcal {S}_i^{(mn)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\big )=T_n^{\, \beta }\big (\omega _n\big (1_{{}_n \mathcal {S}_{i+\beta }^{(mn)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n}\big )\big )=T_n^{\, \beta }(\mathbf {1}^{\mathbf {ss}}_{1,\, \alpha n+mn}) \end{aligned}$$

and we obtain the assertion. Here, $\alpha , \beta \in {\mathbb {Z}}$, $0\le \beta \le n-1$, are such that $d=\alpha n+\beta$. $\square$

Corollary A.7

Let $1\le i\le n-1$ and m be integers and $d\in {\mathbb {Z}}$. Then

$$\begin{aligned} \omega _n\big (1_{{}_n \mathcal {S}_i^{(m n+i)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\big )= \delta _{{}_n \{ i+d \}, 0}\, \upsilon ^{-m-1}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d+m n+i}. \end{aligned}$$

Definition A.8

Let $e\in {\mathbb {Z}}_{\ge 0}$. Define

$$\begin{aligned}&{}_1 \theta _{0, \, e} =\theta _{0, \, e}:=\upsilon ^{e}\, \sum _{\genfrac{}{}{0.0pt}{}{(m_x)_x}{\sum _x\, m_x\deg (x)=e}}\, \prod _{\genfrac{}{}{0.0pt}{}{x\in X}{m_x\ne 0}}\, (1-\upsilon ^{-2\deg (x)})\, 1_{T_x^{(m_x)}},\\&{}_n\theta _{0,\, e} :=\\&{\left\{ \begin{array}{ll} \Omega _n\big (\theta _{0,\, e/n}\big ) &{} \text {for } {}_n \{ e \}=0 ,\\ \displaystyle \upsilon ^{\, {}_n \lfloor e \rfloor +1}\, \sum _{\genfrac{}{}{0.0pt}{}{(m_x)_x}{\sum _x\, m_x\deg (x)= {}_n \lfloor e \rfloor }}\, \prod _{\genfrac{}{}{0.0pt}{}{x\in X{\setminus }\{p\}}{m_x\ne 0}}\, (1-\upsilon ^{-2\deg (x)})\, 1_{\mathcal {T}_x^{(m_x)}}\, (1-\upsilon ^{-2})\, 1_{{}_n \mathcal {S}_{{}_n \{ e \}}^{(m_p n+{}_n \{ e \})}} &{} \text {otherwise} . \end{array}\right. } \end{aligned}$$

$\oslash$

Remark A.9

Note that the definition of $\theta _{0,e}$ as above agrees with the definition in (4.7). $\triangle$

Proposition A.10

Let $d\in {\mathbb {Z}}$. Then

$$\begin{aligned} \omega _n\Big (\sum _{e\in {\mathbb {Z}}_{\ge 0}}\, {}_n\theta _{0,\, e}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\Big )=\sum _{\alpha \in {\mathbb {Z}}_{\ge 0}}\, \xi _\alpha ^{(d)}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d+\alpha }, \end{aligned}$$

where

$$\begin{aligned} \xi _\alpha ^{(d)}:={\left\{ \begin{array}{ll} \displaystyle \xi _{\alpha /n} &{} \text {if } \,\,{}_n \{ d \}=0, \, {}_n \{ \alpha \}=0, \\ \displaystyle \xi _{{}_n \lfloor \alpha \rfloor }-q^{-1}\xi _{{}_n \lfloor \alpha \rfloor }^\circ &{} \text {if } \,\,{}_n \{ d \}\ne 0, \, {}_n \{ \alpha +d \}=0, \\ \displaystyle \xi _{{}_n \lfloor \alpha \rfloor }^\circ &{} \text {if }\,\, {}_n \{ d \}\ne 0, \, {}_n \{ \alpha \}=0,\\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

Here the complex numbers $\xi _s$ and the complex numbers $\xi _s^\circ$, for $s\in {\mathbb {Z}}_{\ge 0}$, are given by

$$\begin{aligned} \xi _{s}&:=\sum _{\genfrac{}{}{0.0pt}{}{(m_x)_{x\in X}, m_x\in {\mathbb {Z}}_{\ge 0}}{\sum _x\,m_x\deg (x)=s}}\, \prod _{\genfrac{}{}{0.0pt}{}{x\in X}{m_x\ne 0}}\, \Big (1-\upsilon ^{-2\deg (x)}\Big ) \quad \text {and} \\ \xi _{s}^\circ&:=\sum _{\genfrac{}{}{0.0pt}{}{(m_x)_{x\in X}, m_x\in {\mathbb {Z}}_{\ge 0}}{m_p=0, \, \sum _x\,m_x\deg (x)=s}}\, \prod _{\genfrac{}{}{0.0pt}{}{x\in X}{m_x\ne 0}}\, \Big (1-\upsilon ^{-2\deg (x)}\Big ). \end{aligned}$$

Proof

Let us first observe that

$$\begin{aligned}&\omega _n\Big (\sum _{e\in {\mathbb {Z}}_{\ge 0}}\, {}_n\theta _{0,\, e}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\Big )\\&\quad =\omega _n\Big (\Big (\sum _{s\ge 0}\, \upsilon ^{\,s}\, \sum _{\genfrac{}{}{0.0pt}{}{(m_x)_x}{m_p=0 ,\ \sum _x\, m_x\deg (x)=s}}\, \prod _{\genfrac{}{}{0.0pt}{}{x\in X{\setminus }\{p\}}{m_x\ne 0}}\, (1-\upsilon ^{-2\deg (x)})\, 1_{\mathcal {T}_x^{(m_x)}}\Big )\\&\qquad \times \, \Big (1+\sum _{t\ge 1}\, \upsilon ^{\, {}_n \lfloor t \rfloor +1-\delta _{{}_n \{ t \},0}}\, (1-\upsilon ^{-2})\, 1_{{}_n \mathcal {S}_{{}_n \{ t \}}^{(t)}}\Big )\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\Big )\Big ). \end{aligned}$$

By applying Corollary A.7, the above quantity is equal to

$$\begin{aligned}&\omega _n\Big (\Big (\sum _{s\ge 0}\, \upsilon ^{\,s}\, \sum _{\genfrac{}{}{0.0pt}{}{(m_x)_x}{m_p=0 ,\ \sum _x\, m_x\deg (x)=s}}\, \prod _{\genfrac{}{}{0.0pt}{}{x\in X{\setminus }\{p\}}{m_x\ne 0}}\, (1-\upsilon ^{-2\deg (x)})\, 1_{\mathcal {T}_x^{(m_x)}}\Big )\\&\quad \times \, ({}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}+(1-\upsilon ^{-2})\, \sum _{t\ge 1}\, \delta _{{}_n \{ d+t \}, 0}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d+t}\Big )\Big ). \end{aligned}$$

Recall that $\omega _n\Big (1_{\mathcal {T}_x^{(m_x)}}\, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\Big )=\upsilon ^{\, -m_x\deg (x)}\, \, {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d+m_x\deg (x)n}$ we get the assertion with

$$\begin{aligned} \xi _\alpha ^{(d)}:={\left\{ \begin{array}{ll} \displaystyle \xi _{{}_n \lfloor \alpha \rfloor }^\circ +(1-\upsilon ^{-2})\, \sum _{\beta =0}^{{}_n \lfloor \alpha \rfloor -1}\,\xi _\beta ^\circ &{} \text {if }\,\, {}_n \{ d \}=0, \, {}_n \{ \alpha \}=0, \\ \displaystyle (1-\upsilon ^{-2})\, \sum _{\beta =0}^{{}_n \lfloor \alpha \rfloor }\,\xi _\beta ^\circ &{} \text {if }\,\, {}_n \{ d \}\ne 0, \, {}_n \{ \alpha +d \}=0, \\ \displaystyle \xi _{{}_n \lfloor \alpha \rfloor }^\circ &{} \text {if }\,\, {}_n \{ d \}\ne 0, \, {}_n \{ \alpha \}=0,\\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

The proposition follows by osserving that

$$\begin{aligned} \xi _{m}^\circ +(1-\upsilon ^{-2})\, \sum _{\beta =0}^{m-1}\,\xi _\beta ^\circ =\xi _{m}. \end{aligned}$$

$\square$

Corollary A.11

(See [51, Corollary 1.4]) As a series in ${\mathbb {C}}[[z]]$, we have

$$\begin{aligned} \sum _{e\ge 0}\, \xi _e\, z^e=\frac{\zeta _X(z)}{\zeta _X(\upsilon ^{-2}\, z)}, \end{aligned}$$

where $\zeta _X(z)$ is the zeta function of X.

Corollary A.12

[36, Lemma 3.14] As a series in ${\mathbb {C}}[[z]]$, we have

$$\begin{aligned} \sum _{e\ge 0}\, \xi _e^\circ \, z^e=\frac{\zeta _{X{\setminus }\{p\}}(z)}{\zeta _{X{\setminus }\{p\}}(\upsilon ^{-2}\, z)}=\frac{\zeta _X(z)}{\zeta _X(\upsilon ^{-2}\, z)}\frac{1-z}{1-\upsilon ^{-2}\, z}, \end{aligned}$$

where $\zeta _X(z)$ and $\zeta _{X{\setminus }\{p\}}(z)$ are the zeta functions of X and $X{\setminus }\{p\}$ respectively.

We conclude this section, by computing the coproduct of the ${}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}$’s: this is a key result to compute the shuffle algebra presentation of $\mathbf {U}_n^>$ in the main body of the paper.

Proposition A.13

Let $d\in {\mathbb {Z}}$. Then

$$\begin{aligned} \tilde{\Delta }\big ({}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\big )={}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d}\otimes 1 + \sum _{e\in {\mathbb {Z}}_{\ge 0}} T_n^{\, {}_n \{ d \}}\big ({}_n\theta _{0,\, e}\big )\,\mathbf {k}_{(1,\, d-e)}\otimes {}_n\mathbf {1}^{\mathbf {ss}}_{1,\, d-e}. \end{aligned}$$

Proof

Set $k:={}_n \lfloor d \rfloor$ and $i:={}_n \{ d \}$. Then

$$\begin{aligned} \tilde{\Delta }\big ({}_n \mathbf {1}^{\mathbf {ss}}_{1,\, kn+i}\big )=\tilde{\Delta }\Big (\sum _{\genfrac{}{}{0.0pt}{}{M\in \mathsf {Pic}(X)}{\deg (M)=k}}\, 1_{\pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i}}\Big ), \end{aligned}$$

and

$$\begin{aligned} \tilde{\Delta }\big (1_{\pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i}}\big )&=\sum _{\mathcal {E}_1,\mathcal {E}_2}\, \upsilon ^{\langle \mathcal {E}_1, \mathcal {E}_2\rangle }\, \mathbf {P}_{\mathcal {E}_1,\mathcal {E}_2}^{\pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i}}\, \frac{\mathbf {a}(\mathcal {E}_1)\mathbf {a}(\mathcal {E}_2)}{\mathbf {a}(\pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i})}1_{\mathcal {E}_1}\, \mathbf {k}_{({\text {rk}}(\mathcal {E}_2), \underline{\deg }(\mathcal {E}_2))}\otimes 1_{\mathcal {E}_2}\\&=\sum _{\mathcal {E}_1\simeq \pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i}}\, 1_{\mathcal {E}_1}\otimes 1\\&\quad +\sum _{{\text {rk}}(\mathcal {E}_1)=0, \mathcal {E}_2\simeq \pi _n^*N\otimes \mathcal {L}_n^{\otimes \, s}}\, \upsilon ^{\langle \mathcal {E}_1, \mathcal {E}_2\rangle }\, \mathbf {P}_{\mathcal {E}_1,\mathcal {E}_2}^{\pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i}}\, \\&\quad \frac{\mathbf {a}(\mathcal {E}_1)\mathbf {a}(\mathcal {E}_2)}{\mathbf {a}(\pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i})}1_{\mathcal {E}_1}\, \mathbf {k}_{(1, \deg (N)n+ s)}\otimes 1_{\mathcal {E}_2}. \end{aligned}$$

Note that

$$\begin{aligned}&\sum _{{\text {rk}}(\mathcal {E}_1)=0, \mathcal {E}_2\simeq \pi _n^*N\otimes \mathcal {L}_n^{\otimes \, s}}\, \upsilon ^{\langle \mathcal {E}_1, \mathcal {E}_2\rangle }\, \mathbf {P}_{\mathcal {E}_1,\mathcal {E}_2}^{\pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i}}\, \frac{\mathbf {a}(\mathcal {E}_1)\mathbf {a}(\mathcal {E}_2)}{\mathbf {a}(\pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i})}1_{\mathcal {E}_1}\, \mathbf {k}_{(1, \deg (N)n+ s)}\otimes 1_{\mathcal {E}_2}\\&\quad =\sum _{{\text {rk}}(\mathcal {F}_1)=0, \mathcal {F}_2\simeq \pi _n^*L\otimes \mathcal {L}_n^{\otimes \, \ell }}\, \upsilon ^{\langle \mathcal {F}_1, \mathcal {F}_2\rangle }\, \mathbf {P}_{\mathcal {F}_1,\mathcal {F}_2}^{\mathcal {O}_{X_n}}\, \mathbf {a}(\mathcal {F}_1) 1_{\mathcal {F}_1\otimes \pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i}}\, \mathbf {k}_{(1,\, \deg (L)n+ \ell +kn+ i)}\\&\qquad \otimes 1_{\mathcal {F}_2\otimes \pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i}}. \end{aligned}$$

Now, $\mathcal {F}_1$ is a torsion sheaf which is a quotient of $\mathcal {O}_{X_n}$, hence it can be only of the form

$$\begin{aligned} \mathcal {F}_1\simeq \bigoplus _{x\in X{\setminus }\{p\}} \mathcal {T}_x^{(m_x)}\oplus {}_n \mathcal {S}_j^{(m_p\, n +j)} \end{aligned}$$

(A.2)

for some $m_x\in {\mathbb {Z}}_{\ge 0}$, with $x\in X$, and some $j\in \{0, \ldots , n-1\}$. Here we formally set $\mathcal {T}_x^{(0)}={}_n \mathcal {S}_n^{(0)}=0$. Moreover, since $\overline{\mathcal {O}_{X_n}}=\overline{\mathcal {F}_1} +\overline{\mathcal {F}_2}$, we get

$$\begin{aligned} \deg (L)&=-\sum _{x\in X}\, m_x\deg (x) -1+\delta _{{}_n \{ j \},0},\\ \ell&={}_n \{ -j \}. \end{aligned}$$

In particular, $\langle \mathcal {F}_1, \mathcal {F}_2\rangle = \deg (L)$.

Moreover, $\mathbf {P}_{\mathcal {F}_1,\mathcal {F}_2}^{\mathcal {O}_{X_n}}\, \mathbf {a}(\mathcal {F}_1)$ is equal to the cardinality of $\mathsf {Hom}^{\mathsf {surj}}(\mathcal {O}_{X_n}, \mathcal {F}_1)$. Since a morphism $\mathcal {O}_{X_n}\rightarrow \mathcal {F}_1$ is surjective if and only if all its components, with respect to the decomposition (A.2) of $\mathcal {F}_1$ are surjective, we need to compute the cardinalities of the space of surjective morphisms from $\mathcal {O}_{X_n}$ to each factor in (A.2). By using the exactness of the functor ${\pi _n}_*$ and the computations of [50, Example 4.12] we have for $m_x, m_p, j\ne 0$

$$\begin{aligned} \# \mathsf {Hom}^{\mathsf {surj}}(\mathcal {O}_{X_n}, \mathcal {T}_x^{(m_x)})&=\# \mathsf {Hom}^{\mathsf {surj}}(\mathcal {O}_{X}, T_x^{(m_x)})=q^{m_x\deg (x)}-q^{(m_x-1)\deg (x)} , \\ \# \mathsf {Hom}^{\mathsf {surj}}(\mathcal {O}_{X_n}, {}_n\mathcal {S}_n^{(m_p\, n)})&=\# \mathsf {Hom}^{\mathsf {surj}}(\mathcal {O}_{X}, T_p^{(m_p)})=q^{m_p}-q^{m_p-1} , \\ \# \mathsf {Hom}^{\mathsf {surj}}(\mathcal {O}_{X_n}, \mathcal {S}_j^{(m_p\, n+j)})&=\# \mathsf {Hom}^{\mathsf {surj}}(\mathcal {O}_{X}, T_p^{(m_p+1)})=q^{m_p+1}-q^{m_p}. \end{aligned}$$

Summarizing, we get

$$\begin{aligned}&\sum _{{\text {rk}}(\mathcal {F}_1)=0, \mathcal {F}_2\simeq \pi _n^*L\otimes \mathcal {L}_n^{\otimes \, \ell }}\, \upsilon ^{\langle \mathcal {F}_1, \mathcal {F}_2\rangle }\, \mathbf {P}_{\mathcal {F}_1,\mathcal {F}_2}^{\mathcal {O}_{X_n}}\, \mathbf {a}(\mathcal {F}_1) 1_{\mathcal {F}_1\otimes \pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i}}\, \mathbf {k}_{(1,\, \deg (L)n+ \ell +kn+ i)}\\&\qquad \otimes \, 1_{\mathcal {F}_2\otimes \pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i}}\\&\quad =\sum _{\genfrac{}{}{0.0pt}{}{\mathcal {F}_1\simeq \bigoplus _{x\in X} \pi _n^*T_x^{(m_x)}}{\mathcal {F}_2\simeq \pi _n^*L}}\, \upsilon ^{-\sum _x\, m_x\deg (x)}\, \prod _{\genfrac{}{}{0.0pt}{}{x\in X}{m_x\ne 0}}\, \big (q^{m_x\deg (x)}-q^{(m_x-1)\deg (x)}\big ) \\&\qquad \times \,1_{\pi _n^*T_x^{(m_x)}}\, \mathbf {k}_{(1,\, d-\sum _x\, m_x\deg (x))}\otimes 1_{\pi _n^*L\otimes \pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i}}\\&\qquad +\,\sum _{j=1}^{n-1}\,\sum _{\genfrac{}{}{0.0pt}{}{\mathcal {F}_1\simeq \bigoplus _{x\in X{\setminus }\{p\}} \mathcal {T}_x^{(m_x)}\oplus {}_n \mathcal {S}_j^{(m_p\, n +j)}}{\mathcal {F}_2\simeq \pi _n^*L\otimes \mathcal {L}_n^{\otimes \, n-j}}}\, \upsilon ^{-\sum _x\, m_x\deg (x)-1}\,\\&\qquad \prod _{\genfrac{}{}{0.0pt}{}{x\in X{\setminus }\{p\}}{m_x\ne 0}}\, \big (q^{m_x\deg (x)}-q^{(m_x-1)\deg (x)}\big )\\&\qquad \times 1_{\pi _n^*T_x^{(m_x)}}\,q\big (q^m-q^{m-1}\big )\, T_n^{\, i}\big (1_{{}_n \mathcal {S}_j^{(m_p n+j)}}\big ) \, \mathbf {k}_{(1,\, d-\sum _x\, m_x\deg (x)-1+n-j)}\\&\qquad \otimes \,1_{\pi _n^*L\otimes \mathcal {L}_n^{\otimes \, n-j}\otimes \pi _n^*M\otimes \mathcal {L}_n^{\otimes \, i}}. \end{aligned}$$

Thus, we get the assertion. $\square$

Remark A.14

For $n=1$, one recovers the non-stacky curve result (cf. [50, Example 4.12])

$$\begin{aligned} \tilde{\Delta }(\mathbf {1}^{\mathbf {ss}}_{1, \, d})=\mathbf {1}^{\mathbf {ss}}_{1,\, d}\otimes 1+\sum _{e\ge 0}\, \theta _{0,\, e}\mathbf {k}_{(1,\, d-e)}\otimes \mathbf {1}^{\mathbf {ss}}_{1,\, d-e}. \end{aligned}$$

$\triangle$

Appendix B: $\mathbf {U}_\upsilon (\mathfrak {sl}(S^1))$ from mirror symmetry

1.1 by Tatsuki Kuwagaki¹²

In the body of this paper, Sala–Schiffmann defined $\mathbf {U}_\upsilon (\mathfrak {sl}(S^1))$ by studying Hall algebras of coherent sheaves over infinite root stacks. In this note, we give a natural explanation of Sala–Schiffmann’s description using mirror symmetry.

1.2 Mirror symmetry for $\mathbb {P}^1_{\infty }$

Let $\mathbb {P}^1_n$ be the projective line over $\mathbb {C}$ with an n-gerbe at $\infty$. Hori–Vafa [26] mirror of $\mathbb {P}^1_n$ is given by a Landau–Ginzburg model $W=z+\frac{1}{z^n}$ over $\mathbb {C}^*$. A traditional category associated with a Landau–Ginzburg model is Fukaya–Seidel category [52]. Here, however, we will use a different description: the partially wrapped Fukaya category. We view $\mathbb {C}^*$ as the cotangent bundle $T^*S^1$ of the circle $S^1=\mathbb {R}/\mathbb {Z}$. Let $p:\mathbb {R}\rightarrow \mathbb {R}/\mathbb {Z}=S^1$ be the quotient map. The Lagrangian skeleton $\Lambda _n$ associated with W, is given by the FLTZ skeleton [18]

$$\begin{aligned} \Lambda _n:=T^*_{S^1}S^1\cup \bigcup _{k\in \mathbb {Z}}T^{*,{<0}}_{p(\frac{k}{n})}S^1 \end{aligned}$$

where $T^*_A S^1$ is the conormal bundle of A inside $S^1$ and $T^{*,<0}_xS^1$ is the negative part of the cotangent fiber at x. We would like to consider the Fukaya category associated with $\Lambda _n$. However there exists a further different description. For a cotangent bundle $T^*X$, the derived category of the infinitesimally wrapped Fukaya category is equivalent to the bounded derived category of $\mathbb {R}$-constructible sheaves over X by Nadler–Zaslow [40, 41]: $\mathfrak {Fuk}(T^*X)\cong {\mathbf{Sh}}^c(X)$. Concerning this result, we can use certain subcategory of ${\mathbf{Sh}}^c(S^1)$ as a model for the Fukaya category associated with $\Lambda _n$ (more precise statement about this consideration is known as Kontsevich’s conjecture. See [21, 32] for further information).

For a sheaf (or a complex of sheaves) $\mathcal {E}$ over a manifold X, Kashiwara–Schapira [30] defined the microsupport ${{\text {SS}}}(\mathcal {E})\subset T^*X$. The definition of “certain subcategory” mentioned above is the full subcategory of ${\mathbf{Sh}}^c(S^1)$ spanned by the objects whose microsupports are contained in $\Lambda _n$. We denote it by ${\mathbf{Sh}}^c_{\Lambda _n}(S^1)$.

In the below, let us change the base field from $\mathbb {C}$ to $k:=\mathbb {F}_q$ i.e. $\mathbb {P}^1_n$ is defined over $\mathbb {F}_q$ and constructible sheaves are $\mathbb {F}_q$-valued. Let $D(\mathsf {Coh}(\mathbb {P}^1_{n}))$ be the bounded derived category of coherent sheaves. We have the following mirror symmetry result:

Theorem B.1

(cf.[18, 19]) There exists an equivalence

$$\begin{aligned} D(\mathsf {Coh}(\mathbb {P}^1_{n}))\simeq {\mathbf{Sh}}^c_{\Lambda _n}(S^1). \end{aligned}$$

Over $\mathbb {C}$, this mirror symmetry is known as “coherent-constructible correspondence” initiated by Bondal, formulated by Fang–Liu–Treumann–Zaslow, proved in full generality by the author [9, 18, 33]. For $\mathbb {P}^1_n$, the statement over $\mathbb {F}_q$ remains true:

Proof of Theorem B.1

There exists a Beilinson collection of $D(\mathsf {Coh}(\mathbb {P}^1_{n}))$ over $\mathbb {F}_q$ [16]. By using the equivalence over $\mathbb {C}$, one can also find the candidate for the exceptional collection on the RHS, which one can verify that it is indeed so also over ${\mathbb {F}}_q$. By calculating the endomorphism algebras of these exceptional collections, one can conclude the equivalence. $\square$

The forgetful morphism $\mathbb {P}^1_{nm}\rightarrow \mathbb {P}^1_n$ induces the pull-back functor $D(\mathsf {Coh}(\mathbb {P}^1_{n}))\rightarrow D(\mathsf {Coh}(\mathbb {P}^1_{nm}))$. On the mirror side, this morphism realized by the inclusion $\Lambda _n\subset \Lambda _{nm}$. We set

$$\begin{aligned} \Lambda _{\infty }:=T^*_{S^1}S^1\cup \bigcup _{q\in \mathbb {Q}}T^{*,{<0}}_{p(q)}S^1 . \end{aligned}$$

Then we have

Corollary B.2

There exists an equivalence

$$\begin{aligned} D(\mathsf {Coh}(\mathbb {P}^1_{\infty }))\simeq {\mathbf{Sh}}^c_{\Lambda _\infty }(S^1). \end{aligned}$$

One can deduce the following corollary easily from the construction of the equivalence above.

Corollary B.3

The abelian category obtained as the essential image of $\mathsf {Coh}_{\mathsf {tors}}(\mathbb {P}^1_\infty )$, the category of torsion sheaves at $\infty$,¹³ in ${\mathbf{Sh}}^c(S^1)$ is the full subcategory spanned by finite direct sums of the following $k^{S^1}_J$ where J is a half-interval of the form $(a,b]\subset \mathbb {R}$ with $a, b\in \mathbb {Q}$ and $k^{S^1}_J:=p_!k_J$.

We denote the category by $\mathsf {Sh}^\mathbb {Q}_{<0}(S^1)$. Below, we use $k_J$ for $p_!k_J^{S^1}$ if the context is clear. Note that our intervals are open-closed but not closed-open as in the body of this paper.

1.3 $\mathbf {U}_\upsilon (\mathfrak {sl}(S^1))$ from the mirror side

The category $\mathsf {Sh}^\mathbb {Q}_{<0}(S^1)$ gives a natural explanation of Sala–Schiffmann’s description Theorem 6.3 as follows.

We give a direct proof of the following (although one can prove by considering Corollary B.2):

Theorem B.4

Let $\mathcal {A}$ be the algebra generated by $\left\{ E_J, K_{J'}^\pm \Big |J, J'\subseteq S^1_{{\mathbb {Q}}}, J\ne S^1_{{\mathbb {Q}}} \right\}$ where J (resp. $J'$) runs over all strict rational open-closed intervals (resp. rational open-closed intervals) modulo the relations appearing in Theorem 6.3. There exists an isomorphism of associative algebras

$$\begin{aligned} \phi :\mathcal {A}\rightarrow {\mathbf {H}}^{\mathsf {tw}}(\mathsf {Sh}^\mathbb {Q}_{<0}(S^1)). \end{aligned}$$

Proof

We define the morphism by the assignment

$$\begin{aligned} E_J\mapsto \upsilon ^{1/2}1_{k_J}, K_J^{\pm 1}\mapsto \mathbf k ^{\pm k_J} \end{aligned}$$

where $1_{k_J}$ is the characteristic function of $k_J$ and $\mathbf k$ is the indefinite of the group ring $\widetilde{\mathbb {Q}}[\mathsf {K}_0^{\mathsf {num}}(\mathsf {Sh}^\mathbb {Q}_{<0}(S^1))]$.

First, we will check the relations in Theorem 6.3 for ${\mathbf {H}}^{\mathsf {tw}}(\mathsf {Sh}^\mathbb {Q}_{<0}(S^1))$.

• (6.1) This is the definition of the product of the twisted Hall algebra; Actually, the description of Euler form given in (1.1) is the Euler form of $\mathsf {Sh}^\mathbb {Q}_{<0}(S^1)$, hence the symmetric one is the same.

• (6.2) For any pair of disjoint $J_1,J_2$ of the form $J_1=(a,b]$ and $J_2=(b,c]$ with $a\ne c$ and $J_1\cup J_2$ is again a proper interval in $S^1_{{\mathbb {Q}}}$, we have an exact sequence

  $$\begin{aligned} 0\rightarrow k_{(b,c]}\rightarrow k_{(a, c]}\rightarrow k_{(a,b]}\rightarrow 0. \end{aligned}$$

  In the Grothendieck group, this gives the desired relation $\mathbf k _{J_1\cup J_2}=\mathbf k _{J_1}+\mathbf k _{J_2}$.

• (6.3) Suppose the same assumption on $J_1, J_2$ as above. The Hall product is

  $$\begin{aligned} \upsilon ^{1/2}1_{k_{J_1}}\cdot \upsilon ^{1/2}1_{k_{J_2}}&=\upsilon (\upsilon ^{-1}1_{k_{J_1\cup J_2}}+\upsilon ^{-1}1_{k_{J_1}+k_{J_2}}),\\ \upsilon ^{1/2}1_{k_{J_2}}\cdot \upsilon ^{1/2}1_{k_{J_1}}&=v1_{k_{J_1}+k_{J_2}}. \end{aligned}$$

  Hence we have

  $$\begin{aligned} \upsilon ^{1/2}(\upsilon ^{1/2}1_{k_{J_1}}\cdot \upsilon ^{1/2}1_{k_{J_2}})-\upsilon ^{-1/2}(\upsilon ^{1/2}1_{k_{J_2}}\cdot \upsilon ^{1/2}1_{k_{J_1}})=\upsilon ^{1/2}1_{k_{J_1\cup J_2}}. \end{aligned}$$

• (6.4) For any $J_1, J_2$ such that $\overline{J_1}\cap \overline{J_2}=\varnothing$. there are no nontrivial extensions $k_{J_1}$ and $k_{J_2}$. This implies that the Hall product is $1_{k_{J_1}}\cdot 1_{k_{J_2}}=1_{k_{J_1}\oplus k_{J_2}}=1_{k_{J_2}}\cdot 1_{k_{J_1}}$. Hence $[1_{k_{J_1}},1_{k_{J_2}}]=0$.

• (6.5) Let $J_1=(a,b]\subsetneq J_2=(c,d]$. First assume that $a \ne c$ and $b\ne d$. There are no nontrivial $\mathsf {Hom}$ and $\mathsf {Ext}$ between $k_{J_1}$ and $k_{J_2}$. The relation (6.5) is trivial. Next we assume that $b=d$, then there exists exists 1-dimensional hom-space $\mathsf {Hom}(k_{J_1}, k_{J_2})$. Hence we have

  $$\begin{aligned} \upsilon ^{\left\langle k_{J_1}, k_{J_2}\right\rangle }\upsilon 1_{k_{J_1}}\cdot 1_{k_{J_2}}&=\upsilon ^2 {1_{k_{J_1}+k_{J_2}}}=q{1_{k_{J_1}+k_{J_2}}},\\ \upsilon ^{\left\langle k_{J_2}, k_{J_1}\right\rangle }\upsilon 1_{k_{J_2}}\cdot 1_{k_{J_1}}&=q{1_{k_{J_1}+k_{J_2}}}. \end{aligned}$$

  Since $\left\langle k_{J_1}, k_{J_2}\right\rangle =\left\langle \chi _{J_1}, \chi _{{J}_2}\right\rangle$ where the intervals are interpreted as closed-open intervals on the RHS, we get (6.5) for this case. Similarly, one can prove the case of $a=c$.

It is obvious that the morphism $\phi$ is surjective. One can follow the argument for the injectivity of Theorem 6.3. $\square$

We can readapt the definitions of the coproduct (6.7) and of the Green pairing (6.8) in our setting endowing ${\mathbf {H}}^{\mathsf {tw}}(\mathsf {Sh}^\mathbb {Q}_{<0}(S^1))$ with the structure of a topological $\widetilde{{\mathbb {Q}}}$-Hopf algebra. By passing to the reduced Drinfeld double, we obtain the following.

Corollary B.5

We have an isomorphism

$$\begin{aligned} \mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{{\mathbb {Q}}}))\cong \mathbf {D}({\mathbf {H}}^{\mathsf {tw}}(\mathsf {Sh}^\mathbb {Q}_{<0}(S^1))). \end{aligned}$$

In the above argument, the rationality of $S^1_{{\mathbb {Q}}}$ is not essential. We set

$$\begin{aligned} \Lambda _{\mathbb {R}}:=T^*_{S^1}S^1\cup \bigcup _{q\in \mathbb {R}}T^{*,{<0}}_{p(q)}S^1. \end{aligned}$$

Let $\mathsf {Sh}^\mathbb {R}_{<0}(S^1)$ be the abelian full subcategory of ${\mathbf{Sh}}^c_{\Lambda _\mathbb {R}}(S^1)$ spanned by finite direct sums of elements $k_J$, where J is an open-closed interval of the form $(a,b]\subset \mathbb {R}$ and $k_J:=p_!k_J$ by abuse of notation.

Theorem B.6

Let $\mathcal {A}$ be the algebra generated by $\left\{ E_J, K_{J'}^\pm \Big |J, J'\subseteq S^1, J\ne S^1\right\}$ where J (resp. $J'$) runs over all strict open-closed intervals (resp. open-closed intervals) modulo the relations appearing in Theorem 6.3. There exists an isomorphism

$$\begin{aligned} \phi :\mathcal {A}\rightarrow {\mathbf {H}}^{\mathsf {tw}}(\mathsf {Sh}^\mathbb {R}_{<0}(S^1)). \end{aligned}$$

Proof

We define the morphism by the assignment

$$\begin{aligned} E_J\mapsto \upsilon ^{1/2}1_{k_J}, K_J^{\pm 1}\mapsto \mathbf k ^{\pm k_J}. \end{aligned}$$

One can prove that this is a homomorphism by the same argument as in the proof of Theorem B.4. The surjectivity is obvious.

For the injectivity, we follow the argument of Theorem 6.3. Let $P\subset S^1$ be a finite subset. Let us consider the subalgebra $\mathcal {A}_P\subset \mathcal {A}$ generated by $F_J, K_J^{\pm 1}$ for J ends in P. Since $\mathcal {A}_P\cong \mathcal {A}_{\vert P\vert }$ in the proof of Theorem 6.3, the restriction $\phi |_{\mathcal {A}_P}$ is injective. Since $\bigcup _{P}\mathcal {A}_P=\mathcal {A}$, we complete the proof. $\square$

As before, we can suitably define a coproduct and a Green pairing giving rise to the structure of a topological $\widetilde{{\mathbb {Q}}}$-Hopf algebra on ${\mathbf {H}}^{\mathsf {tw}}(\mathsf {Sh}^\mathbb {R}_{<0}(S^1))$. By taking the reduced Drinfeld double, we obtain.

Corollary B.7

We have an isomorphism

$$\begin{aligned} \mathbf {U}_\upsilon (\mathfrak {sl}(S^1))\cong \mathbf {D}({\mathbf {H}}^{\mathsf {tw}}(\mathsf {Sh}^\mathbb {R}_{<0}(S^1))). \end{aligned}$$

1.4 Fundamental representation

Along with Sala–Schiffmann’s definition for $\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{{\mathbb {Q}}}))$, the fundamental representation of $\mathbf {U}_\upsilon (\mathfrak {sl}(S^1))$ is defined by the $\widetilde{\mathbb {Q}}$-vector space, $\mathbb {V}_{S^1}:=\bigoplus _{y\in \mathbb {R}}\widetilde{\mathbb {Q}}\vec {u}_y$ with action given by

$$\begin{aligned} \begin{aligned} F_{(a,b]}\bullet \vec {u}_y&:=\delta _{\{b+y\},0}\upsilon ^{1/2}\vec {u}_{y+b-a},\\ E_{(a,b]}\bullet \vec {u}_y&:=\delta _{\{a+y\},0}\upsilon ^{-1/2}\vec {u}_{y+a-b},\\ K^\pm _{(a,b]}\bullet \vec {u}_y&:=\upsilon ^{\pm (\delta _{\{b+y\},0}-\delta _{\{a+y\},0})}\vec {u}_{y}. \end{aligned} \end{aligned}$$

(B.1)

Next we define ${\mathbf {U}}_{\infty }^>[r]$. Let us set

$$\begin{aligned} {\mathbf {1}}^{\mathsf {ss}}_{x}&:=1_{p_!k_{[0,x]}} \text { if }x\ge 0,\\ {\mathbf {1}}^{\mathsf {ss}}_{x}&:=1_{p_!k_{(x,0)}[1]} \text { if }x< 0, \end{aligned}$$

for $x\in \mathbb {R}$. Let ${\mathbf {U}}_\infty ^>$ be the Hall algebra generated by ${\mathbf {1}}^\mathsf {ss}_x$’s. Let us denote by ${\mathbf {U}}^>_\infty [r]$ the linear span of products of r generators ${\mathbf {1}}^\mathsf {ss}_x$. We call the number r the rank.

The following is an analog of Sala–Schiffmann’s result:

Theorem B.8

The natural Hecke action of $\mathbf {U}_\upsilon (\mathfrak {sl}(S^1))$ on ${\mathbf {U}}^>_\infty [1]$ is isomorphic to $\mathbb {V}_{S^1}$, after applying the automorphism (6.12).

Proof

There exists a basis $\{{\mathbf {1}}^\mathsf {ss}_x\}_{x\in \mathbb {R}}$ of ${\mathbf {U}}^>_\infty [1]$. This gives a natural identification with $\mathbb {V}_{S^1}$ as vector spaces by $\vec {u}_y\mapsto {\mathbf {1}}_{-y}^{\mathsf {ss}}$. The second and third formulas in (B.1) follow directly by computations. The first one is followed by the definition of Drinfeld double. $\square$

1.5 The composition Hall algebra of $S^1$

Recall the definition of $\mathbf {C}_1$ (cf. Sect. 4.2.2), which is generated by

$$\begin{aligned} {\mathbf {1}}_{0,\,d}:=\sum _{\genfrac{}{}{0.0pt}{}{\mathcal {T}\in \mathsf {Tor}(\mathbb {P}^1)}{deg(\mathcal {T})=d}} {1}_\mathcal {T}. \end{aligned}$$

In the language of the constructible side, a torsion sheaf corresponds to a finite direct sum of the following:

• (1) k-local system on $S^1$,

• (2) $p_!k_{[0,n)}$ for some $n\in \mathbb {Z}_{>0}$,

• (3) $p_!k_{(0,n]}$ for some $n\in \mathbb {Z}_{\ge 0}$.

We call such an object a torsion object. For a torsion object $E=L\oplus \bigoplus _{i=1}^Ip_!k_{[0,n_i)} \oplus \bigoplus _{j=1}^Jp_!k_{(0,m_j]}$, we set $\deg E:={\text {rk}}L+\sum _{i=1}^In_i+\sum _{j=1}^Jm_j$. Then

$$\begin{aligned} {\mathbf {1}}'_{0,\,d}:=\sum _{\genfrac{}{}{0.0pt}{}{E\text { : torsion}}{\deg (E)=d}} {1}_E. \end{aligned}$$

is the mirror of ${\mathbf {1}}_{0,d}$.

Sala–Schiffmann defined $\mathbf {U}^0_\infty$ as the subalgebra of $\mathbf {H}_\infty ^\mathsf {tw}(X)$ generated by $\mathbf {C}_\infty =\mathbf {U}_\upsilon ^+(\mathfrak {sl}(S^1_{\mathbb {Q}}))$ and the pull-back of $\mathbf {C}$. By imitating this definition, we obtain:

Definition B.9

The composition Hall algebra $\mathbf {U}^0_\mathbb {R}$ is an algebra generated by $\mathbf {H}^{\mathsf {tw}}(\mathsf {Sh}^\mathbb {Q}_{<0}(S^1))$ and $\{{\mathbf {1}}'_{0,\,d}\}_{d\le 0}$. $\oslash$

1.6 Other Dynkin types

In this section, we treat other Dynkin types: $A_n, D_n, \hat{D}_n$.

1.6.1 $A_n$

Set $I=[0,1]$. Consider the Lagrangian

$$\begin{aligned} \Lambda _{A_n}:=T^*_II\cup \bigcup _{i=0}^nT_{i/n}^{*,<0}I \end{aligned}$$

and the category of constructible sheaves microsupported on ${\mathbf{Sh}}^c_{\Lambda _{A_n}}(I)$. The full subcategory $\mathsf {Sh}_{\Lambda _{A_n}}(I)$ consisting of constant sheaves supported on open-closed intervals of the forms like (i / n, j / n] forms an abelian category and is equivalent to the category of representations of the quiver $A_n$. The reduced Drinfeld double of the twisted Hall algebra of $\mathsf {Sh}_{\Lambda _{A_n}}(I)$ is isomorphic to the quantum group $\mathbf {U}_\upsilon (\mathfrak {sl}(n+1))$ of type $A_n$.

We define two subsets:

$$\begin{aligned} \Lambda _{A_\mathbb {Q}}&:=T^*_II\cup \bigcup _{a\in I\cap \mathbb {Q}}T_{a}^{*,<0}I\\ \Lambda _{A_\mathbb {R}}&:=T^*_II\cup \bigcup _{a\in I }T_{a}^{*,<0}I \end{aligned}$$

Then one can define the full subcategory $\mathsf {Sh}_{\Lambda _{A_\mathbb {Q}}}(I)$ (resp. $\mathsf {Sh}_{\Lambda _{A_{\mathbb {R}}}}(I)$) consisting of constant sheaves supported on open-closed intervals of the forms like (a, b] with $a,b\in I\cap \mathbb {Q}$ (resp. $a,b\in I$) forms a finitary abelian category. We define

$$\begin{aligned} \mathbf {U}_\upsilon (\mathfrak {sl}(I_{{\mathbb {Q}}}))&:=\mathbf {D}(\mathbf {H}^{\mathsf {tw}} (\mathsf {Sh}_{\Lambda _{A_\mathbb {Q}}}(I))),\\ \mathbf {U}_\upsilon (\mathfrak {sl}(I))&:=\mathbf {D}(\mathbf {H}^{\mathsf {tw}} (\mathsf {Sh}_{\Lambda _{A_\mathbb {R}}}(I))). \end{aligned}$$

1.6.2 $D_n, \widehat{D}_n$

Since one can easily imagine $\widehat{D}_n$ case from $D_n$, we only treat the latter one.

Consider the following subset in $\mathbb {R}^2$:

$$\begin{aligned} I:=\left\{ (x, 0)\Big |x\in [0,1]\right\} \cup \left\{ (x, e^{1/x})\Big |-1< x< 0\right\} \cup \left\{ (x, -e^{1/x})\Big |-1< x< 0\right\} . \end{aligned}$$

(B.2)

We call the components of this union $I_1, I_2, I_3$ from left to right.

$$\begin{aligned} \Lambda _{D_n}:=\bigcup _{i=1}^3T^*_{I_i}\mathbb {R}^2 \cup \bigcup _{i=0}^{n-3}T^{*,<0}_{(i/(n-2),0)}\mathbb {R}^2 \end{aligned}$$

Here $T^{*,<0}_{(x,y)}\mathbb {R}^2$ is the subset of $T^*_{(x,y)}\mathbb {R}^2=\{(x, y; \xi , \eta )\}$ ($\xi$ (resp. $\eta$) is the cotangent coordinate of x (resp. y)) defined by $\xi <0$.

Note that $I_1\cup I_2$ and $I_1\cup I_3$ are smooth curves isomorphic to closed intervals. A closed-open interval in I is an interval of the form (a, b] of either $I_1\cup I_2$ or $I_1\cup I_3$. Let D be the closed disk with diameter 1 in $\mathbb {R}^2$. Let $\mathsf {Sh}_{\Lambda _{D_n}}(D)\subset {\mathbf{Sh}}^c_{\Lambda _{D_n}}(D)$ be the full abelian subcategory consisting of constant sheaves on open-closed intervals. One can easily see that this abelian category is equivalent to the category of representations of the quiver $D_n$. Hence the reduced Drinfeld double of the twisted Hall algebra of $\mathsf {Sh}_{\Lambda _{D_n}}(D)$ is isomorphic to the quantum group of type $D_n$.

We define two isotropic sets:

$$\begin{aligned} \Lambda _{D_\mathbb {Q}}&:=\bigcup _{i=1}^3T^*_{I_i}\mathbb {R}^2\cup \bigcup _{a\in I_1\cap \mathbb {Q}}T^{*,<0}_{a}\mathbb {R}^2,\\ \Lambda _{D_\mathbb {R}}&:=\bigcup _{i=1}^3T^*_{I_i}\mathbb {R}^2 \cup \bigcup _{a\in I_1}T^{*,<0}_{a}\mathbb {R}^2. \end{aligned}$$

Then one can define the full subcategory $\mathsf {Sh}_{\Lambda _{D_\mathbb {Q}}}(D)$ (resp. $\mathsf {Sh}_{\Lambda _{D_{\mathbb {R}}}}(D)$) consisting of constant sheaves supported on closed-open intervals with their ends in $\Lambda _{D_\mathbb {Q}}$ (resp, $\Lambda _{D_{\mathbb {R}}}$) forms a finitary abelian category. We define

$$\begin{aligned} \mathbf {U}_\upsilon (\mathfrak {so}(2I_{{\mathbb {Q}}}))&:=\mathbf {D}(\mathbf {H}^{\text {tw}} (\mathsf {Sh}_{\Lambda _{D_\mathbb {Q}}}(D))),\\ \mathbf {U}_\upsilon (\mathfrak {so}(2I))&:=\mathbf {D}(\mathbf {H}^{\text {tw}} (\mathsf {Sh}_{\Lambda _{D_\mathbb {R}}}(D))). \end{aligned}$$

To make things more explicit, let us write up the Euler forms here. There are three new situations which did not appear in $A_n$-case. Let $a, b\in (0,1]$.

• Case (T):
  $$\begin{aligned} H^\bullet {\mathbb {R}}\mathsf {Hom}(k_{\overline{I_2} \cup (0, a]}, k_{\overline{I_3}})\simeq 0\quad \text{ and }\quad H^\bullet {\mathbb {R}}\mathsf {Hom}(k_{\overline{I_3}}, k_{\overline{I_2}\cup (0, a]})\simeq k[-1]. \end{aligned}$$
• Case (Y):
  $$\begin{aligned} H^\bullet \, {\mathbb {R}}\mathsf {Hom}(k_{\overline{I_2}\cup (0, a]}, k_{\overline{I_3}\cup (0,b]})\simeq {\left\{ \begin{array}{ll} k &{} \text {if }\,\,a> b, \\ 0 &{} \text {otherwise} . \end{array}\right. } \end{aligned}$$
• Case (V):
  $$\begin{aligned} H^\bullet {\mathbb {R}}\mathsf {Hom}(k_{\overline{I_2}}, k_{\overline{I_3}})\simeq 0. \end{aligned}$$

Remark B.10

One can also consider the subset (B.2) of ${\mathbb {R}}^2$ with the opposite orientation, that is,

$$\begin{aligned} I:=\left\{ (x, 0)\Big |x\in [-1, 0]\right\} \cup \left\{ (x, e^{1/x})\Big |0< x\le 1\right\} \cup \left\{ (x, -e^{1/x})\Big |0<x\le 1\right\} . \end{aligned}$$

We call the components of this union $I_1, I_2, I_3$ from left to right. In such a case the formulas above may be rewritten as follows, for $a, b\in [-1, 0)$:

Case (T$^{\prime }$)::

$$\begin{aligned} H^\bullet \, {\mathbb {R}}\mathsf {Hom}(k_{I_2\cup (a, 0]}, k_{I_3})\simeq k[-1]\quad \text{ and }\quad H^\bullet \, {\mathbb {R}}\mathsf {Hom}(k_{I_3}, k_{I_2\cup (a, 0]})\simeq 0. \end{aligned}$$

Case (Y$^{\prime }$)::

$$\begin{aligned} H^\bullet {\mathbb {R}}\mathsf {Hom}(k_{I_2\cup (a, 0]}, k_{I_3\cup (b,0]})\simeq {\left\{ \begin{array}{ll} k&{} \text {if }\,\,a< b, \\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

Case (V$^{\prime }$)::

$$\begin{aligned} H^\bullet \, {\mathbb {R}}\mathsf {Hom}(k_{I_2}, k_{I_3})\simeq 0. \end{aligned}$$

$\triangle$