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Borel and shifted category O

David Hernandez, Andrei Neguţ

Abstract

We prove a precise relation between simple modules in the Borel category O and the shifted category O for a symmetrizable Kac-Moody Lie algebra.

Borel and shifted category O

Abstract

We prove a precise relation between simple modules in the Borel category O and the shifted category O for a symmetrizable Kac-Moody Lie algebra.
Paper Structure (30 sections, 16 theorems, 232 equations, 1 figure)

This paper contains 30 sections, 16 theorems, 232 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Borel category ${\mathcal{O}}$
  3. The shifted category ${\mathcal{O}}^{\text{sh}}$
  4. $QQ$-systems
  5. Shuffle algebras
  6. Acknowledgements
  7. The Borel category ${\mathcal{O}}$
  8. Basic notations
  9. The quantum affine algebra
  10. The quantum loop algebra
  11. Kac-Moody Lie algebras
  12. The big shuffle algebra
  13. Hopf algebras
  14. The (small) shuffle algebra
  15. Slope subalgebras and category ${\mathcal{O}}$
  16. ...and 15 more sections

Key Result

Theorem 1.1

For any symmetrizable Kac-Moody Lie algebra ${\mathfrak{g}}$ and any rational $\ell$-weight ${\boldsymbol{\psi}}$, we have a vector space isomorphism which preserves the natural gradings by ${\boldsymbol{n}} \in {\mathbb{N}^I}$ and $\boldsymbol{x} \in ({\mathbb{C}}^*)^{{\boldsymbol{n}}}$ on both sides (see eqn:decomposition x and eqn:decomposition x shifted). Therefore, eqn:main intro descends to

Figures (1)

  • Figure 1: A distinguished zig-zag $Z$

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Example 2.8
  • ...and 36 more