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The Alekseev-Meinrenken diffeomorphism arising from the Stokes phenomenon

Xiaomeng Xu

Abstract

The Alekseev-Meinrenken diffeomorphism is a distinguished diffeomorphism from the space of $n\times n$ Hermitian matrices to the space of $n\times n$ positive definite Hermitian matrices. This paper derives the explicit expression of the diffeomorphism, via the Stokes phenomenon of meromorphic linear systems of ordinary differential equations with Poncaré rank $1$.

The Alekseev-Meinrenken diffeomorphism arising from the Stokes phenomenon

Abstract

The Alekseev-Meinrenken diffeomorphism is a distinguished diffeomorphism from the space of Hermitian matrices to the space of positive definite Hermitian matrices. This paper derives the explicit expression of the diffeomorphism, via the Stokes phenomenon of meromorphic linear systems of ordinary differential equations with Poncaré rank .
Paper Structure (17 sections, 21 theorems, 95 equations)

This paper contains 17 sections, 21 theorems, 95 equations.

Table of Contents

  1. Introduction and the main result
  2. Stokes phenomenon and monodromy data
  3. Canonical solutions
  4. Stokes matrices
  5. Isomonodromy deformation
  6. The boundary value and explicit Stokes matrices
  7. Explicit Riemann-Hilbert map at a caterpillar point
  8. The explicit Alekseev-Meinrenken diffeomorphism
  9. The Gelfand-Tsetlin coordinates
  10. Riemann-Hilbert-Birkhoff maps at caterpillar points are compatible with Gelfand-Tsetlin systems
  11. Explicit formula of the Alekseev-Meinrenken diffeomorphism
  12. The Alekseev-Meinrenken diffeomorphim
  13. Principal bundles and cross sections
  14. Phase transformations
  15. Proof of Theorem \ref{['exAM']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

AM There exists a unique diffeomorphism which intertwines the Gelfand-Tsetlin systems on both sides (and has one extra property, see Section AMdiff). In particular, the map $\Gamma_{AM}$ is a Poisson isomorphism.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • ...and 21 more