Jordan-Hölder property for shifted quantum affine algebras

TL;DR

The paper resolves the Jordan–Hölder stability question for finite-length representations in the category $\mathcal{O}^{\mathrm{sh}}$ of shifted quantum affine algebras, showing closure under fusion products and that simples descend to truncations. It builds a comprehensive truncation framework (simply-connected, intermediate, adjoint) and connects representation-theoretic data to universal R-matrix polynomiality, enabling explicit descent criteria and finite-length conclusions. These results yield subring structures in the Grothendieck group, illuminate links to cluster algebras, and substantiate Langlands dual $q$-character interpretations for simple modules. The work thus provides a rigorous bridge between shifted quantum affine representation theory, truncation theory, and cluster-algebraic structures, with potential implications for quantized Coulomb branches and Langlands duality in this setting.

Abstract

We prove that finite length representations of shifted quantum affine algebras in category $\mathcal{O}^{\mathrm{sh}}$ are stable by fusion product. This implies that in the topological Grothendieck ring $K_0(\mathcal{O}^{\mathrm{sh}})$ the Grothendieck group of finite length representations forms a non-topological subring. We also conjecture this subring is isomorphic to the cluster algebra discovered in arXiv:2401.04616. In the course of our proofs, we establish that any simple representation in category $\mathcal{O}^{\mathrm{sh}}$ descends to a truncation, for certain truncation parameters as conjectured in arXiv:2010.06996 in terms of Langlands dual $q$-characters.

Theorems & Definitions (51)

• Proposition 2.1
• Remark 2.2
• Proposition 2.3
• Definition 2.4
• Example 2.5
• Theorem 2.6
• Theorem 2.7
• Remark 3.1
• Proposition 3.2
• proof