Quantum cluster algebras and representations of shifted quantum affine algebras

2025/07/07 by Paganelli, Francesca · 2 citations
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2507.05008

Abstract

We construct a new quantization K_t(O^sh_ℤ) of the Grothendieck ring of the category O^sh_ℤ of representations of shifted quantum affine algebras (of simply-laced type). We establish that our quantization is compatible with the quantum Grothendieck ring K_t(O^\mathfrakb,+_ℤ) for the quantum Borel affine algebra, namely that there is a natural embedding K_t(O^\mathfrakb,+_ℤ)\hookrightarrow K_t(O^sh_ℤ). Our construction is partially based on the cluster algebra structure on the classical Grothendieck ring discovered by Geiss-Hernandez-Leclerc. As first applications, we formulate a quantum analogue of QQ-systems (that we make completely explicit in type A₁). We also prove that the quantum oscillator algebra is isomorphic to a localization of a subalgebra of our quantum Grothendieck ring and that it is also isomorphic to the Berenstein-Zelevinsky's quantum double Bruhat cell ℂ_t[SL₂^w₀,w₀].

Citations

• Representations of shifted affine quantum groups and Coulomb branches
• Based cluster algebras of infinite rank
• Isomorphisms among quantum Grothendieck rings and cluster algebras
• Weyl group symmetry of q-characters
• Monoidal categorification and quantum affine algebras II
• A quantum cluster algebra approach to representations of simply-laced quantum affine algebras
• The Berenstein-Zelevinsky quantum cluster algebra conjecture
• Quantum cluster algebras
• Cluster algebras III: Upper bounds and double Bruhat cells
• Braid Group Actions and Tensor Products
• Quiver varieties and t-analogs of q-characters of quantum affine algebras
• Perverse sheaves and quantum Grothendieck rings
• The q-characters of representations of quantum affine algebras and deformations of W-algebras
• Double Bruhat cells and total positivity

Cited by

• Representations and characters of quantum affine algebras at the crossroads between cluster categorification and quantum integrable models
• An introduction to representation-theoretic canonical bases of cluster algebras

Related

• A quantum cluster algebra approach to representations of simply-laced quantum affine algebras
• Quantum affine algebras and cluster algebras
• Toroidal Grothendieck rings and cluster algebras
• Real simple modules over simply-laced quantum affine algebras and categorifications of cluster algebras
• Quantization of virtual Grothendieck rings and their structure including quantum cluster algebras