Characteristic equation for symplectic groupoid and cluster algebras

TL;DR

This paper derives a characteristic equation for symplectic groupoids related to cluster algebras, expressing roots in terms of Casimir elements, and extends the framework to a symplectic case, with implications for quantum versions.

Contribution

It introduces a new expression for roots of the characteristic equation using Casimir elements in symplectic groupoids linked to cluster algebras, including quantum generalizations.

Findings

Roots of the characteristic equation are simple monomials of cluster Casimir elements.

The results hold in both classical and quantum cases.

Generalization to a symplectic groupoid is established.

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 Teichm\"uller spaces for the algebra $sl_n$. 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 $\mathcal A_{Sp_{2m}}$-groupoid.