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Affine highest weight structures on module categories over quiver Hecke algebras

Haruto Murata

Abstract

We prove that the category of finitely generated graded modules over the quiver Hecke algebra of arbitrary type admits numerous stratifications in the sense of Kleshchev. A direct consequence is that the full subcategory corresponding to the quantum unipotent subgroup associated with any Weyl group element is an affine highest weight category. Our results significantly generalize earlier works by Kato, Brundan, Kleshchev, McNamara and Muth. The key ingredient is a realization of standard modules via determinantial modules. We utilize the technique of R-matrices to study these standard modules.

Affine highest weight structures on module categories over quiver Hecke algebras

Abstract

We prove that the category of finitely generated graded modules over the quiver Hecke algebra of arbitrary type admits numerous stratifications in the sense of Kleshchev. A direct consequence is that the full subcategory corresponding to the quantum unipotent subgroup associated with any Weyl group element is an affine highest weight category. Our results significantly generalize earlier works by Kato, Brundan, Kleshchev, McNamara and Muth. The key ingredient is a realization of standard modules via determinantial modules. We utilize the technique of R-matrices to study these standard modules.

Paper Structure

This paper contains 35 sections, 115 theorems, 292 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and convention
  4. Laurentian algebras
  5. Quantum groups
  6. Global bases and quantum minors
  7. Quiver Hecke algebras
  8. Cyclotomic quiver Hecke algebras
  9. R-matrices
  10. Unmixing sequences
  11. Cuspidal decomposition
  12. Determinantial modules
  13. Full subcategories corresponding to quantum unipotent subgroups
  14. Properties of affinizations of determinantial modules
  15. Homological property of $\widehat{M}(w\Lambda_i, \Lambda_i)$
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $\beta \in Q_+$. The category ${R(\beta)}\textup{-gMod}$ of finitely generated graded $R(\beta)$-modules is stratified with standard modules $\Delta(\lambda,s) \ (\sum_{1 \leq k \leq l} \lambda_k\beta_k + \beta_s = \beta)$. Precisely speaking, there exists a certain preorder $\leq$ on the labell

Figures (1)

  • Figure 1: Affine rank 2 positive roots

Theorems & Definitions (216)

  • Theorem 1.1: Theorem \ref{['thm:standard']}, Theorem \ref{['thm:stratification']}
  • Theorem 1.2: Theorem \ref{['thm:affinehighest']}
  • Corollary 1.3: Corollary \ref{['cor:gldim']}
  • Theorem 1.4: Theorem \ref{['thm:ext1']}
  • Proposition 1.5: Corollary \ref{['cor:M1ext']}
  • Proposition 1.6: Theorem \ref{['thm:SES']}
  • Theorem 1.7: Proposition \ref{['prop:BGGformula']}
  • Lemma 2.1: MR3495746
  • Lemma 2.4
  • proof
  • ...and 206 more