Characteristic equation for symplectic groupoid and cluster algebras

2021/01/14 by Leonid Chekhov, Chekhov, Leonid O., Michael Shapiro +3
Mathematics · #51H25 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2101.10323

openalex publication_date 2021/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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-λ\mathbb A^T)=0, with \mathbb A∈ \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 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.

Citations

• Total positivity, Grassmannians, and networks
• Isomonodromic deformations and twisted Yangians arising in Teichmüller theory
• Matching polytopes, toric geometry, and the non-negative part of the Grassmannian

Related

• Darboux coordinates for symplectic groupoid and cluster algebras
• Symplectic groupoid and cluster algebras
• Geometric leaf of symplectic groupoid
• Cluster algebras and Weil-Petersson forms
• Cluster Algebras, Symplectic Leaves and Quantum Groups