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Using Fock--Goncharov higher Teichmüller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the A_n groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric R-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for A_3 and A_4 in terms of geodesic functions for Riemann surfaces with holes. We represent braid-group transformations for A_n via sequences of cluster mutations in the special A_n-quiver. We prove the groupoid relations for quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit.
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ROOTS OF CHARACTERISTIC EQUATION FOR SYMPLECTIC GROUPOID
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 A_n-groupoid of upper-triangular matrices to express roots of the characteristic equation det(A−λA^T)=0, with A∈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 A_n-groupoid to a A_{Sp_2m}-groupoid.
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- PAR ID:
- 10345549
- Date Published:
- Journal Name:
- Uspehi matematičeskih nauk
- Volume:
- 77
- Issue:
- 3
- ISSN:
- 0042-1316
- Page Range / eLocation ID:
- 177-178
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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