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Brief Communications
Roots of the characteristic equation for the symplectic groupoid
L. O. Chekhovabc, M. Z. Shapirocb, H. Shibod a Steklov Mathematical Institute of Russian Academy of Sciences
b National Research University Higher School of Economics
c Michigan State University, East Lansing, USA
d Xi'an Jiaotong University, Xi'an, Shaanxi, P.R. China
Received: 17.03.2022
Morphisms on the set An⊆gln of unipotent upper-triangular n×n matrices are transformations A↦BAB⊤∈An. For B=eεg linearized transformations δεA=gA+Ag⊤ are analogous to Krichever–Novikov transformations [4]. The quantization of the Bondal Poisson structure on An (see [1]) obtained using the symplectic groupoid construction is the reflection equation with trigonometric R-matrix
Rn(q)1ARt1n(q)2A=2ARt1n(q)1ARn(q).
Here A is an upper-triangular metric with diagonal entries q−1/2 and self-adjoint operator entries ai,j for i<j. Then the combination AA−†:=A[A†]−1 undergoes an adjoint transformation AA−†↦BAA−†B−1, and we are to find its eigenvalues λi∈C determining [n/2] independent Casimir elements [1].
We express the entries of A using the quantum Fock–Goncharov variables Zα=Z(i,j,k) parameterized in terms of the barycentric coordinates (i+j+k=n) of the vertices of the bn-quiver [3] (Fig. 1, left): a solid arrow from α to β means that ZβZα=q−2ZαZβ, and a dashed arrow gives ZβZα=q−1ZαZβ. The directed network N dual to the bn-quiver is the directed graph of double arrows in Fig. 1.
With any oriented path P:j⇝ in any planar directed network \mathcal N we associate the quantum weight w(P), which is the Weyl ordered (denoted by the symbol ; see [2]) product of the variables Z_\alpha of all faces of \mathcal N lying to the right of the path.
In the case of the b_n-quiver we define the three quantum (n\times n)-transport matrices
\begin{equation*}
\begin{gathered} \, (\mathcal M_1)_{i,j}= \sum_{\substack{P\colon j\rightsquigarrow i'}} w(P),\quad (\mathcal M_2)_{i,j}= \sum_{\substack{P\colon j\rightsquigarrow i'' }} w(P), \quad (\mathcal M_3)_{i,j}= \sum_{\substack{P\colon j' \rightsquigarrow i'' }} w(P), \end{gathered}
\end{equation*}
\notag
where the paths contributing to \mathcal M_3 are obtained by reversing all horizontal double arrows in N. Note that {\mathcal M}_1 and \mathcal M_3 are lower-triangular matrices and {\mathcal M}_2 is an upper-triangular matrix. We define an antidiagonal matrix
\begin{equation*}
S=\sum_{i=1}^n (-1)^{i+1}q^{-i+1/2}e_{i,n+1-i}
\end{equation*}
\notag
and let denote the products of Fock–Goncharov variables along SE-diagonals of the b_n-quiver.
We have the groupoid condition [2] {\mathcal M}_3 S {\mathcal M}_1= {\mathcal M}_2, and Theorem 4.1 of [2] states that \mathbb A:=\mathcal M_1^\top\mathcal M_3 S \mathcal M_1 satisfies equation (1). Amalgamating the variables Z_{(i,0,n-i)} and Z_{(0,n-i,i)} pairwise we obtain the new Casimirs used to eliminate the variables Z_{(l,n-l,0)}, so that we obtain the \mathcal A_n-quiver (Fig. 1, right), whose Casimirs are the [n/2] elements for 1\leqslant i<n/2 and C_{n/2}=T_{n/2} for integer n/2.
Theorem. The eigenvalues \lambda_i\in \mathbb C, 1\leqslant i\leqslant n, of the operator \mathbb A\mathbb A^{-\unicode{8224}} are
\begin{equation*}
\lambda_i=(-1)^{n-1} q^{-n}\times \begin{cases} \prod\limits_{k=i}^{[n/2]} C_k&\textit{for}\ 1\leqslant i\leqslant [n/2]; \\ 1\hphantom{\sum\limits^\sum} &\textit{for } i=(n+1)/2 \textit{ for odd } n; \\ \prod\limits_{k=n+1-i}^{[n/2]}C_k^{-1}&\textit{for}\ n-[n/2]+1\leqslant i \leqslant n. \end{cases}
\end{equation*}
\notag
Proof. For \mathbb A=\mathcal M_1^\top\!\mathcal M_3 S \mathcal M_1 we have \mathcal M_i^\unicode{8224}=\mathcal M_i^\top and \mathbb A^{\unicode{8224}}\!= \mathcal M_1^\top S^+\! \mathcal M_3^\top \mathcal M_1, so that \mathbb A-\lambda \mathbb A^{\unicode{8224}}= \mathcal M_1^\top\bigl(\mathcal M_3S- \lambda S^+ \mathcal M_3^\top\bigr)\mathcal M_1, and the singularity equation becomes \bigl(\mathcal M_3S- \lambda S^+ \mathcal M_3^\top\bigr)\psi=0. The matrix \mathcal M_3 is lower triangular and its diagonal entries are m_1=Z_{(n,0,0)} and , 2\leqslant i \leqslant n. The crucial observation is that both the matrices \mathcal M_3S and S^+\mathcal M_3^\top are upper antitriangular. The antidiagonal components of these matrices are
\begin{equation*}
\sum_{i=1}^n (-1)^{i+1}q^{i-1/2} m_i e_{n+1-i,i}\quad \text{and}\quad \sum_{i=1}^n (-1)^{n-i}q^{-n+i-1/2} m_{n+1-i} e_{n+1-i,i},
\end{equation*}
\notag
respectively, and the singularity equation has a non-trivial solution if a combination of them contains the zero element, so that the admissible values are \lambda_i=(-1)^{n-1}q^{-n} m_{n+1-i}/m_i; the quotients of variables correspond to different cases in the theorem. \Box
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Bibliography
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A. I. Bondal, Izv. Ross. Akad. Nauk Ser. Mat., 68:4 (2004), 19–74 ; English transl. in Izv. Math., 68:4 (2004), 659–708 |
2. |
L. O. Chekhov and M. Shapiro, “Log-canonical coordinates for symplectic groupoid and cluster algebras”, IMRN (to appear) |
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V. Fock and A. Goncharov, Publ. Math. Inst. Hautes Études Sci., 103 (2006), 1–211 |
4. |
I. M. Krichever and S. P. Novikov, Uspekhi Mat. Nauk, 54:6(330) (1999), 149–150 ; English transl. in Russian Math. Surv., 54:6 (1999), 1248–1249 |
Citation:
L. O. Chekhov, M. Z. Shapiro, H. Shibo, “Roots of the characteristic equation for the symplectic groupoid”, Russian Math. Surveys, 77:3 (2022), 552–554
Linking options:
https://www.mathnet.ru/eng/rm9999https://doi.org/10.1070/RM9999 https://www.mathnet.ru/eng/rm/v77/i3/p177
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Abstract page: | 330 | Russian version PDF: | 37 | English version PDF: | 47 | Russian version HTML: | 146 | English version HTML: | 112 | References: | 59 | First page: | 14 |
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