# Darboux coordinates for symplectic groupoid and cluster algebras

@inproceedings{LChekhov2020DarbouxCF, title={Darboux coordinates for symplectic groupoid and cluster algebras}, author={L.Chekhov and M.Shapiro}, year={2020} }

Using Fock–Goncharov higher Teichmüller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the An groupoid of uppertriangular 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 A3 and A4 in terms…

## 10 Citations

### Characteristic equation for symplectic groupoid and cluster algebras

- Mathematics
- 2021

. 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…

### Stokes Manifolds and Cluster Algebras

- MathematicsCommunications in Mathematical Physics
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Stokes’ manifolds, also known as wild character varieties, carry a natural Poisson structure. Our goal is to provide explicit log-canonical coordinates for this Poisson structure on the Stokes’…

### Cluster variables for affine Lie--Poisson systems

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We show that having any planar (cyclic or acyclic) directed network on a disc with the only condition that all n1+m sources are separated from all n2+m sinks, we can construct a cluster-algebra…

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It is explicitly shown that the Poisson bracket on the set of shear coordinates defined by V. V. Fock in 1997 induces the Fenchel–Nielsen bracket on the set of gluing parameters (length and twist…

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- 2021

We generalize Bonahon and Wong's $\mathrm{SL}_2(\mathbb{C})$-quantum trace map to the setting of $\mathrm{SL}_3(\mathbb{C})$. More precisely, for each non-zero complex number $q$, we associate to…

### Wilson lines and their Laurent positivity.

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- 2020

For a marked surface $\Sigma$ and a semisimple algebraic group $G$ of adjoint type, we study the Wilson line function $g_{[c]}:\mathcal{P}_{G,\Sigma} \to G$ associated with the homotopy class of an…

### The ∗-Markov Equation for Laurent Polynomials

- MathematicsMoscow Mathematical Journal
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We consider the $*$-Markov equation for the symmetric Laurent polynomials in three variables with integer coefficients, which is an equivariant analog of the classical Markov equation for integers.…

### POINTS OF QUANTUM SLn COMING FROM QUANTUM SNAKES

- Mathematics
- 2021

We show that the quantized Fock-Goncharov monodromy matrices satisfy the relations of the quantum special linear group SLn. The proof employs a quantum version of the technology invented by…

### Points of quantum $\mathrm{SL}_n$ coming from quantum snakes

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We show that the Fock-Goncharov quantum monodromy matrices satisfy the relations of the quantum special linear group SLn. The proof employs a quantum version of the technology invented by Fock and…

### Noncommutative Networks on a Cylinder.

- Mathematics
- 2020

In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder $\Sigma$,…

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