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BGG Reciprocity for Generalized Weight Modules of Unrolled Restricted Quantum Groups at Roots of Unity

Matthew Rupert

Abstract

We consider the category of generalized weight modules over the unrolled restricted quantum group $\overline{U}_q^H(\mathfrak{g})$ of a finite-dimensional simple complex Lie algebra $\mathfrak{g}$ at root of unity q. Although this category does not admit projective modules, it is filtered by subcategories with enough projectives. We show that the projective modules in these subcategories satisfy a variant of BGG reciprocity and give a generators and relations description of the projective covers of irreducible modules when $\mathfrak{g}=\mathfrak{sl}_2$.

BGG Reciprocity for Generalized Weight Modules of Unrolled Restricted Quantum Groups at Roots of Unity

Abstract

We consider the category of generalized weight modules over the unrolled restricted quantum group of a finite-dimensional simple complex Lie algebra at root of unity q. Although this category does not admit projective modules, it is filtered by subcategories with enough projectives. We show that the projective modules in these subcategories satisfy a variant of BGG reciprocity and give a generators and relations description of the projective covers of irreducible modules when .
Paper Structure (6 sections, 19 theorems, 95 equations)

This paper contains 6 sections, 19 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Preliminaries
  4. Generalized Weight Modules
  5. Projective Modules
  6. Projective Covers for Irreducible $\overline{U}_q^{H}(\mathfrak{sl}_2)$-modules.

Key Result

Theorem 1.1

$\mathcal{C}_{\bar{m}}$ has enough projectives.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 24 more