Characteristic equation for symplectic groupoid and cluster algebras

Abstract

We use the Darboux coordinate representation found by two of the authors (L.Ch. and M.Sh.) for entries of general symplectic leaves of the ${\mathcal{A}}_n$-groupoid of upper-triangular matrices to express roots of the characteristic equation $\det (\mathbb{A}-\lambda {\mathbb{A}}^{\text{T}})=0$, with $\mathbb{A}\in {\mathcal{A}}_n$, in terms of Casimirs of this Darboux coordinate representation, which is based on cluster variables of Fock--Goncharov higher Teichmüller spaces for the algebra $s{l}_n$. We show that roots of the characteristic equation are simple monomials of cluster Casimir elements. This statement remains valid in the quantum case as well. We consider a generalization of ${\mathcal{A}}_n$-groupoid to a ${\mathcal{A}}_{S{p}_{2m}}$-groupoid.

15 pages, 7 figures, section on $S{p}_{2m}$ groupoid added. arXiv admin note: text overlap with arXiv:2003.07499

References in corpus (5)

• Total positivity, Grassmannians, and networks
• Matching polytopes, toric geometry, and the non-negative part of the Grassmannian
• Continuous tensor categories from quantum groups I: algebraic aspects
• Isomonodromic deformations and twisted Yangians arising in Teichmüller theory
• On a Poisson space of bilinear forms with a Poisson Lie action