• Corpus ID: 212736892

Darboux coordinates for symplectic groupoid and cluster algebras

  title={Darboux coordinates for symplectic groupoid and cluster algebras},
  author={L.Chekhov and M.Shapiro},
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… 

Characteristic equation for symplectic groupoid and cluster algebras

. 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

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Fenchel–Nielsen coordinates and Goldman brackets

  • L. Chekhov
  • Mathematics
    Russian Mathematical Surveys
  • 2020
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

Quantum traces for $\mathrm{SL}_n(\mathbb{C})$: the case $n=3$.

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

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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

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.


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

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Isomonodromic deformations and twisted Yangians arising in Teichm

Isomonodromic deformations and twisted Yangiansarising in Teichmüller theory

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Coisotropic calculus and Poisson groupoids

Lagrangian submanif olds play a special role in the geometry of symplectic manifolds. From the point of view of quantization theory, or simply a categorical approach to symplectic geometry [Gu-S2],


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We point out, and draw some consequences of, the fact that the Poisson Lie group G* dual to G=GL_n(C) (with its standard complex Poisson structure) may be identified with a certain moduli space of

Homotopy groups and (2 + 1)-dimensional quantum de Sitter gravity

Cluster Algebras and Poisson Geometry

Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in