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Deformed Cartan matrices and generalized preprojective algebras II: General type

Ryo Fujita, Kota Murakami

Abstract

We propose a definition of deformed symmetrizable generalized Cartan matrices with several deformation parameters, which admit a categorical interpretation by graded modules over the generalized preprojective algebras in the sense of Geiß-Leclerc-Schröer. Using the categorical interpretation, we deduce a combinatorial formula for the inverses of our deformed Cartan matrices in terms of braid group actions. Under a certain condition, which is satisfied in all the symmetric cases or in all the finite and affine cases, our definition coincides with that of the mass-deformed Cartan matrices introduced by Kimura-Pestun in their study of quiver $\mathcal{W}$-algebras.

Deformed Cartan matrices and generalized preprojective algebras II: General type

Abstract

We propose a definition of deformed symmetrizable generalized Cartan matrices with several deformation parameters, which admit a categorical interpretation by graded modules over the generalized preprojective algebras in the sense of Geiß-Leclerc-Schröer. Using the categorical interpretation, we deduce a combinatorial formula for the inverses of our deformed Cartan matrices in terms of braid group actions. Under a certain condition, which is satisfied in all the symmetric cases or in all the finite and affine cases, our definition coincides with that of the mass-deformed Cartan matrices introduced by Kimura-Pestun in their study of quiver -algebras.
Paper Structure (17 sections, 29 theorems, 92 equations)

This paper contains 17 sections, 29 theorems, 92 equations.

Table of Contents

  1. Deformed Cartan matrices
  2. Notation
  3. Deformed Cartan matrices
  4. Braid group actions
  5. Remark on finite type
  6. Combinatorial inversion formulas
  7. Generalized preprojective algebras
  8. Conventions
  9. Preliminary on positively graded algebras
  10. Generalized preprojective algebras
  11. Projective resolutions
  12. Euler-Poincaré pairing
  13. Braid group action
  14. Remarks
  15. Comparison with Kimura-Pestun's deformation
  16. ...and 2 more sections

Key Result

Theorem 1.3

When $C$ is of infinite type, the matrix $\widetilde{C}(q,t,\ul{\mu})$ has non-negative coefficients, namely we have $\widetilde{C}_{ij}(q,t,\ul{\mu}) \in \mathbb{Z}_{\ge 0}[\Gamma_0][\![t]\!]$ for any $i,j \in I$.

Theorems & Definitions (71)

  • Remark 1.1
  • Example 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 1.5
  • Lemma 1.6
  • proof
  • Proposition 1.7
  • proof
  • Theorem 1.8
  • ...and 61 more