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Categorification of integrable representations of quantum groups

Hao Zheng

Abstract

We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac-Moody algebra. As byproducts, we obtain a geometric realization of Lusztig's canonical bases of these representations as well as a new positivity result. The main ingredient in the underlying geometric construction is a class of micro-local perverse sheaves on quiver varieties.

Categorification of integrable representations of quantum groups

Abstract

We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac-Moody algebra. As byproducts, we obtain a geometric realization of Lusztig's canonical bases of these representations as well as a new positivity result. The main ingredient in the underlying geometric construction is a class of micro-local perverse sheaves on quiver varieties.

Paper Structure

This paper contains 34 sections, 28 theorems, 172 equations.

Table of Contents

  1. Preliminaries
  2. Quantum groups
  3. Perverse sheaves on algebraic stacks
  4. Decomposition theorem
  5. Fourier-Deligne transform
  6. Fourier inversion formula
  7. Localization in triangulated categories
  8. Example
  9. Categories and functors from quiver varieties
  10. The category $\mathfrak{D}$
  11. Notations
  12. Quiver varieties
  13. Fourier-Deligne transform
  14. Localization
  15. A micro-local point of view
  16. ...and 19 more sections

Key Result

Proposition 2.1.5

There is an isomorphism of functors $\Phi_{\Omega_2,\Omega_3}\Phi_{\Omega_1,\Omega_2}\cong\Phi_{\Omega_1,\Omega_3}$ for orientations $\Omega_1,\Omega_2,\Omega_3\subset\hat{H}$.

Theorems & Definitions (33)

  • Remark 1.1.3
  • Remark 1.4.2
  • Proposition 2.1.5
  • Remark 2.3.2
  • Proposition 2.3.3
  • Proposition 2.3.4
  • Remark 2.3.9
  • Proposition 2.4.3
  • Proposition 2.4.4
  • Proposition 2.5.1
  • ...and 23 more