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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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This content will become publicly available on January 1, 2026
GT-shadows for the gentle version \widehat{GT}_{gen} of the Grothendieck-Teichmueller group
Let $$\B_3$$ be the Artin braid group on $$3$$ strands and $$\PB_3$$ be the corresponding pure braid group. In this paper, we construct the groupoid $$\GTSh$$ of $$\GT$$-shadows for a (possibly more tractable) version $$\GTh_0$$ of the Grothendieck-Teichmueller group $$\GTh$$ introduced in paper \cite{HS-fund-groups} by D. Harbater and L. Schneps. We call this group the gentle version of $$\GTh$$ and denote it by $$\GTh_{gen}$$. The objects of $$\GTSh$$ are finite index normal subgroups $$\N$$ of $$\B_3$$ satisfying the condition $$\N \le \PB_3$$. Morphisms of $$\GTSh$$ are called $$\GT$$-shadows and they may be thought of as approximations to elements of $$\GTh_{gen}$$. We show how $$\GT$$-shadows can be obtained from elements of $$\GTh_{gen}$$ and prove that $$\GTh_{gen}$$ is isomorphic to the limit of a certain functor defined in terms of the groupoid $$\GTSh$$. Using this result, we get a criterion for identifying genuine $$\GT$$-shadows.
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- Award ID(s):
- 1745583
- PAR ID:
- 10614905
- Publisher / Repository:
- Elsevier BV
- Date Published:
- Journal Name:
- Journal of Pure and Applied Algebra
- Volume:
- 229
- Issue:
- 1
- ISSN:
- 0022-4049
- Page Range / eLocation ID:
- 107819
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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