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Freezing operators in representation theory of quantum loop algebras

Ryo Fujita, Fan Qin

Abstract

We prove the Hernandez conjecture on the simple $(q,t)$-characters (an analog of the Kazhdan--Lusztig conjecture) for untwisted quantum loop algebras of classical type. This result is new in type $\mathrm{C}$. We also prove that the folding homomorphism, introduced by Hernandez, gives a dimension-preserving bijective correspondence between the finite-dimensional simple representations (in a skeletal subcategory) of untwisted quantum loop algebras of classical simply-laced type and those of the corresponding doubly-twisted quantum loop algebras. This result is new in type $\mathrm{D}$. In our approach, we develop a bootstrapping method for $q$ and $(q,t)$-characters, based on the freezing operator previously introduced in the context of cluster algebras by the second named author. This method allows us to reduce statements for general simple representations in all classical types to corresponding results on core subcategories in a uniform manner.

Freezing operators in representation theory of quantum loop algebras

Abstract

We prove the Hernandez conjecture on the simple -characters (an analog of the Kazhdan--Lusztig conjecture) for untwisted quantum loop algebras of classical type. This result is new in type . We also prove that the folding homomorphism, introduced by Hernandez, gives a dimension-preserving bijective correspondence between the finite-dimensional simple representations (in a skeletal subcategory) of untwisted quantum loop algebras of classical simply-laced type and those of the corresponding doubly-twisted quantum loop algebras. This result is new in type . In our approach, we develop a bootstrapping method for and -characters, based on the freezing operator previously introduced in the context of cluster algebras by the second named author. This method allows us to reduce statements for general simple representations in all classical types to corresponding results on core subcategories in a uniform manner.
Paper Structure (30 sections, 25 theorems, 77 equations)

This paper contains 30 sections, 25 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main result
  4. Application to twisted quantum loop algebras
  5. Organization
  6. Simple $(q,t)$-characters and Hernandez's conjecture
  7. Notation
  8. Quantum loop algebras
  9. Finite-dimensional simple modules
  10. The $q$-characters
  11. Skeletal subcategory $\mathscr{C}_{\mathbb Z}$
  12. Quantum Grothendieck rings
  13. Simple $(q,t)$-characters
  14. Hernandez's conjecture
  15. Freezing operator
  16. ...and 15 more sections

Key Result

Proposition 2.2

The map $[M] \mapsto \chi_q(M)$ gives rise to an injective ring homomorphism In particular, the ring $K(\mathscr{C})$ is commutative. As a commutative ring, $K(\mathscr{C})$ is freely generated by the classes of fundamental modules.

Theorems & Definitions (45)

  • Example 2.1
  • Proposition 2.2: FR
  • Theorem 2.3: FM
  • Proposition 2.4
  • proof
  • Theorem 2.5: Her, see also HO
  • Proposition 2.6: Nakajima04Her
  • Remark 2.7
  • Theorem 2.8: Nakajima04
  • Conjecture 2.9: Her
  • ...and 35 more