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Russian Mathematical Surveys, 2022, Volume 77, Issue 3, Pages 552–554
DOI: https://doi.org/10.1070/RM9999
(Mi rm9999)
 

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
References:
Funding agency Grant number
National Science Foundation DMS-1702115
Ministry of Science and Higher Education of the Russian Federation 075-15-2021-608
M. S. was supported by NSF grant DMS-1702115; L. Ch. and M. S. were supported by the International Laboratory of Cluster Geometry of HSE University, RF Government grant (ag. no. 075-15-2021-608 of 08.JUN.21). H. Sh. thanks the Department of Mathematics of Michigan State University and the exchange program “Discover America” for their support and stimulating research atmosphere during his visit to MSU where the main part of the work was accomplished.
Received: 17.03.2022
Bibliographic databases:
Document Type: Article
MSC: Primary 15A24; Secondary 16T20
Language: English
Original paper language: Russian

Morphisms on the set Angln of unipotent upper-triangular n×n matrices are transformations ABABAn. 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 q1/2 and self-adjoint operator entries ai,j for i<j. Then the combination AA:=A[A]1 undergoes an adjoint transformation AABAAB1, and we are to find its eigenvalues λiC 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=q2ZαZβ, and a dashed arrow gives ZβZα=q1ZαZβ. 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

Bibliography

1. A. I. Bondal, Izv. Ross. Akad. Nauk Ser. Mat., 68:4 (2004), 19–74  mathnet  crossref  mathscinet  zmath; English transl. in Izv. Math., 68:4 (2004), 659–708  crossref
2. L. O. Chekhov and M. Shapiro, “Log-canonical coordinates for symplectic groupoid and cluster algebras”, IMRN (to appear)  crossref
3. V. Fock and A. Goncharov, Publ. Math. Inst. Hautes Études Sci., 103 (2006), 1–211  crossref  mathscinet  zmath
4. I. M. Krichever and S. P. Novikov, Uspekhi Mat. Nauk, 54:6(330) (1999), 149–150  mathnet  crossref  mathscinet  zmath; English transl. in Russian Math. Surv., 54:6 (1999), 1248–1249  crossref  adsnasa

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
Citation in format AMSBIB
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\by L.~O.~Chekhov, M.~Z.~Shapiro, H.~Shibo
\paper Roots of the characteristic equation for the symplectic groupoid
\jour Russian Math. Surveys
\yr 2022
\vol 77
\issue 3
\pages 552--554
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\crossref{https://doi.org/10.1070/RM9999}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4461381}
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