Stokes Matrices and Poisson Lie Groups

Abstract

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 meromorphic connections on the unit disc having an irregular singularity at the origin. The Riemann-Hilbert map for such connections, taking the Stokes data, induces a holomorphic map from the dual of the Lie algebra of G to the Poisson Lie group G*. The main result is that this map is Poisson. First this leads to new, more direct, proofs of theorems of Duistermaat and Ginzburg-Weinstein (enabling one to reduce Kostant's non-linear convexity theorem, involving the Iwasawa projection, to the linear convexity theorem, involving the `diagonal part'). Secondly we obtain a new approach to the braid group invariant Poisson structure on Dubrovin's local moduli space of semisimple Frobenius manifolds: it is induced from the standard Poisson structure on G*.

23 pages, 1 figure, (top margin adjusted)

References in corpus (2)

• Geometry and analytic theory of Frobenius manifolds
• On a Poisson structure on the space of Stokes matrices

Cited by in corpus (35)

• Quasi-Hamiltonian Geometry of Meromorphic Connections
• Stability conditions and Stokes factors
• Geometry of Multiplicative Preprojective Algebra
• Wild nonabelian Hodge theory on curves
• Symmetries and invariants of twisted quantum algebras and associated Poisson algebras
• Calabi-Yau structures on topological Fukaya categories
• Dynamics on Wild Character Varieties
• Poisson involutions, spin Calogero-Moser systems associated with symmetric Lie subalgebras and the symmetric space spin Ruijsenaars-Schneider models
• Quasi-Coxeter algebras, Dynkin diagram cohomology and quantum Weyl groups
• Poisson groups and differential Galois theory of Schroedinger equation on the circle
• A Lie-theoretic description of the solution space of the tt*-Toda equations
• Geometry of moduli spaces of meromorphic connections on curves, Stokes data, wild nonabelian Hodge theory, hyperkahler manifolds, isomonodromic deformations, Painleve equations, and relations to Lie theory
• G-bundles, isomonodromy and quantum Weyl groups
• Moduli spaces of irregular singular connections
• Regularized limits of Stokes matrices, isomonodromy deformation and crystal basis
• The $U(n)$ Gelfand-Zeitlin system as a tropical limit of Ginzburg-Weinstein diffeomorphisms
• Representations of quantum groups arising from the Stokes phenomenon
• Stokes phenomenon and Yang-Baxter equations
• Comparison of Poisson structures and Poisson-Lie dynamical r-matrices
• An explicit isomorphism between quantum and classical sl(n)
• A rigid irregular connection on the projective line
• Riemann-Hilbert-Birkhoff inverse problem for semisimple flat $F$-manifolds, and convergence of oriented associativity potentials
• Isomonodromy aspects of the tt* equations of Cecotti and Vafa II. Riemann-Hilbert problem
• Frobenius manifolds and quantum groups
• Stability data, irregular connections and tropical curves
• Stokes manifolds and cluster algebras
• Ginzburg-Weinstein via Gelfand-Zeitlin
• Kostant, Steinberg, and the Stokes matrices of the tt*-Toda equations
• Euler continuants in noncommutative quasi-Poisson geometry
• Stokes phenomenon, Gelfand-Zeitlin systems and relative Ginzburg-Weinstein linearization
• WKB approximation, crystals and combinatorics of Young tableaux
• A quantum duality principle for coisotropic subgroups and Poisson quotients
• Irregular Riemann-Hilbert correspondence, Alekseev-Meinrenken dynamical r-matrices and Drinfeld twists
• Polystability of Stokes representations and differential Galois groups
• Morita equivalences, moduli spaces and flag varieties