On minimal affinizations of representations of quantum groups

Abstract

In this paper we study minimal affinizations of representations of quantum groups (generalizations of Kirillov-Reshetikhin modules of quantum affine algebras introduced by Chari). We prove that all minimal affinizations in types A, B, G are special in the sense of monomials. Although this property is not satisfied in general, we also prove an analog property for a large class of minimal affinization in types C, D, F. As an application, the Frenkel-Mukhin algorithm works for these modules. For minimal affinizations of type A, B we prove the thin property (the l-weight spaces are of dimension 1) and a conjecture of Nakai-Nakanishi (already known for type A). The proof of the special property is extended uniformly for more general quantum affinizations of quantum Kac-Moody algebras.

38 pages; references and additional results added. Accepted for publication in Communications in Mathematical Physics

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Cited by in corpus (29)

• Cluster algebras and quantum affine algebras
• Baxter's Relations and Spectra of Quantum Integrable Models
• Spectra of quantum KdV Hamiltonians, Langlands duality, and affine opers
• Path description of type B q-characters
• Simple tensor products
• Cluster algebras and category O for representations of Borel subalgebras of quantum affine algebras
• Langlands duality for finite-dimensional representations of quantum affine algebras
• Extended T-systems
• Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
• Isomorphisms among quantum Grothendieck rings and propagation of positivity
• Minimal affinizations as projective objects
• Graded Limits of Minimal Affinizations in Type D
• M-systems and Cluster algebras
• Graded limits of minimal affinizations over the quantum loop algebra of type $G_2$
• Jacobi-Trudi determinants and characters of minimal affinizations
• Extended $T$-System of Type $G_2$
• Three-vertex prime graphs and reality of trees
• Dorey's Rule and the q-Characters of Simply-Laced Quantum Affine Algebras
• On the extended T-system of type $C_3$
• A functor for constructing $R$-matrices in the category $\mathcal{O}$ of Borel quantum loop algebras
• On Tensor Products of a Minimal Affinization with an Extreme Kirillov-Reshetikhin Module for type A
• Paths and tableaux descriptions of Jacobi-Trudi determinant associated with quantum affine algebra of type D_n
• On Frenkel-Mukhin algorithm for q-character of quantum affine algebras
• On q,t-characters and the l-weight Jordan filtration of standard quantum affine sl2 modules
• Tensor products and $q$-characters of HL-modules and monoidal categorifications
• Non-Minimality of Certain Irregular Coherent Preminimal Affinizations
• On the minimal affinizations of type $F_4$
• Quantum periodicity and Kirillov-Reshetikhin modules
• Inflations for representations of shifted quantum affine algebras