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Representations of quantum groups arising from the Stokes phenomenon

Xiaomeng Xu

Abstract

In this paper we prove that the quantum Stokes matrices of the quantum differential equation at a second order pole give rise to representations of the quantum group $U_q(\frak{gl}_n)$. We explain our results from the viewpoint of deformation quantization of the classical Stokes matrices at a second order pole. As a consequence, we can get a dictionary between the theory of Stokes phenomenon and the theory of quantum groups. We briefly discuss several such correspondences, and outline the generalization of our results to all classical types of Lie algebras and to the quantum differential equation at an arbitrary order pole.

Representations of quantum groups arising from the Stokes phenomenon

Abstract

In this paper we prove that the quantum Stokes matrices of the quantum differential equation at a second order pole give rise to representations of the quantum group . We explain our results from the viewpoint of deformation quantization of the classical Stokes matrices at a second order pole. As a consequence, we can get a dictionary between the theory of Stokes phenomenon and the theory of quantum groups. We briefly discuss several such correspondences, and outline the generalization of our results to all classical types of Lie algebras and to the quantum differential equation at an arbitrary order pole.

Paper Structure

This paper contains 35 sections, 44 theorems, 226 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Motivation: deformation quantization of the classical Stokes matrices at second order poles
  4. The main application of Theorem \ref{['introthm1']}: crystal basis arising from the WKB approximation
  5. Outlook
  6. Stokes matrices of the equation d
  7. The unique formal fundamental solution of equation \ref{['introqeq']} in nonresonant case
  8. The canonical solutions with a prescribed asymptotics
  9. Stokes matrices
  10. Stokes matrices satisfy the RLL relation
  11. Solutions of the Knizhnik–Zamolodchikov equations with irregular singularities
  12. The unique formal solution of equation \ref{['gcKZ1']} in the nonresonant case
  13. Anti-Stokes rays and canonical solutions of the equation \ref{['gcKZ1']} with prescribed asymptotics
  14. Solutions of the system of equations \ref{['gcKZ1']} and \ref{['gcKZ2']} on different domains
  15. Stokes factors
  16. ...and 20 more sections

Key Result

Theorem 1.1

(c.f. Theorem thm1) For any fixed $h\notin\mathbb{Q}$ and $u\in\mathfrak h_{\rm reg}$, the map (with $q=e^{\pi\mathrm{i} h}$) defines a representation of the Drinfeld-Jimbo quantum group $U_q(\frak{gl}_n)$ on the vector space $L(\lambda)$. Here recall that $U_q(\frak{gl}_n)$ is a unital associative algebra with generators $q^{\pm h_i}, e_j, f_j,$$1\le j\le n-1, 1\le i\le n$ and relations:

Figures (8)

  • Figure 1: The anti-Stokes rays of the equation \ref{['gcKZ1']}. The solid lines are the fixed rays (independent of $t$) labelled by $d_0,...,d_{2l-1}, d_{2l}=d_0$. The dashed lines are the rays varying with respect to $t$.
  • Figure 2: The configuration of the anti-Stokes rays as $t$ close to $0$ (or $\infty$). The solid lines are the fixed rays (independent of $t$), and two adjacent such rays are labelled by $d_i$ and $d_{i+1}$. The dashed lines, the rays varying with respect to $t$, are close to the solid lines as $t$ near $0$ (or $\infty$). And $d(t), d'(t)$ represent the two adjacent rays approaching to $d_i$ and $d_{i+1}$ as $t\rightarrow 0$ (or $\infty$) respectively.
  • Figure 3: Transposition of $t$ near $1$ such that $t$ passes above $1$.
  • Figure 4: Transposition of $t$ near $0$ such that $t$ passes above $0$.
  • Figure 5: Transposition of $t$ near $\infty$ such that $t$ passes below $0$ and $1$.
  • ...and 3 more figures

Theorems & Definitions (91)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Proposition 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 81 more