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Monodromy of the Casimir connection of a symmetrisable Kac-Moody algebra

Andrea Appel, Valerio Toledano-Laredo

Abstract

Let g be a symmetrisable Kac-Moody algebra and V an integrable g-module in category O. We show that the monodromy of the (normally ordered) rational Casimir connection on V can be made equivariant with respect to the Weyl group W of g, and therefore defines an action of the braid group B_W of W on V. We then prove that this action is canonically equivalent to the quantum Weyl group action of B_W on a quantum deformation of V, that is an integrable, category O-module V_h over the quantum group U_h(g) such that V_h/hV_h is isomorphic to V. This extends a result of the second author which is valid for g semisimple.

Monodromy of the Casimir connection of a symmetrisable Kac-Moody algebra

Abstract

Let g be a symmetrisable Kac-Moody algebra and V an integrable g-module in category O. We show that the monodromy of the (normally ordered) rational Casimir connection on V can be made equivariant with respect to the Weyl group W of g, and therefore defines an action of the braid group B_W of W on V. We then prove that this action is canonically equivalent to the quantum Weyl group action of B_W on a quantum deformation of V, that is an integrable, category O-module V_h over the quantum group U_h(g) such that V_h/hV_h is isomorphic to V. This extends a result of the second author which is valid for g semisimple.

Paper Structure

This paper contains 211 sections, 83 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Acknowledgments
  4. The Casimir connection
  5. Kac--Moody algebras
  6. Realisations of matrices
  7. Kac--Moody algebras
  8. Symmetrisable Kac--Moody algebras
  9. Diagrammatic Kac--Moody algebras
  10. The Casimir connection
  11. Fundamental group of root system arrangements
  12. The Casimir connection
  13. Local finiteness
  14. Flatness
  15. Proof
  16. ...and 196 more sections

Key Result

Theorem 1

The ($W$--equivariant) monodromy of $\nabla_{:\mathcal{K}:}$ on a category $\mathcal{O}$ integrable $\mathfrak{g}$--module $V$ is canonically equivalent to the quantum Weyl group action of the braid group $\mathcal{B}_{W}$ on a quantum deformation of $V$.

Theorems & Definitions (105)

  • Theorem
  • Lemma
  • Definition
  • Theorem
  • Definition
  • Theorem
  • Definition
  • Definition
  • Lemma
  • Proposition
  • ...and 95 more