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Canonical basis, KLR-algebras and parity sheaves

Ruslan Maksimau

Abstract

We give a construction of a basis of the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated with a Dynkin quiver in terms of parity sheaves.

Canonical basis, KLR-algebras and parity sheaves

Abstract

We give a construction of a basis of the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated with a Dynkin quiver in terms of parity sheaves.

Paper Structure

This paper contains 36 sections, 54 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Geometric construction of KLR-algebras
  3. Quivers
  4. KLR-algebra
  5. A faithful representation of $R_{\nu,A}$
  6. Reminder on Borel-Moore homology
  7. Algebra $\mathbf{Z}_{V,A}$
  8. Algebras $\mathbf{P}_{V,A}$ and $\mathbf{S}_{V,A}$
  9. Stratification on $F_V\times F_V$
  10. Stratification on $Z_{V}$
  11. Generators of $\mathbf{Z}_{V,A}$
  12. The $A$-algebra structure on $\mathbf{F}_{V,A}$
  13. KLR-algebra and Borel-Moore homology
  14. Stratified varieties
  15. Lusztig category $\mathcal{Q}_V$
  16. ...and 21 more sections

Key Result

Theorem 1.1

Theorems & Definitions (107)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Theorem 2.4
  • proof
  • ...and 97 more