Characterization of the unit object in localized quantum unipotent category

2025/11/13 by Matsuura, Koh, Nakashima, Toshiki
#Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2511.10157

Abstract

For the quiver Hecke algebra R, let R\hbox-gmod be the category of finite-dimensional graded R-modules, and let \widetildeR\hbox-gmod[w] be the localization of R\hbox-gmod. Kashiwara and the second author showed the set of equivalence classes of simple objects up to grading shifts Irr(\widetildeR\hbox-gmod[w]) in \widetildeR\hbox-gmod[w] has a crystal structure, and Irr(\widetildeR\hbox-gmod[w]) is isomorphic to the so-called cellular crystal \mathbb B_{\mathbf i}. This isomorphism induces a function ε_i^* on \mathbb B_{\mathbf i}. We give an explicit formula of ε_i^*, and using this formula, we give a characterization of the unit object of \widetildeR\hbox-gmod[w] for the case of classical finite types.

Related

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• Categorified Crystal Structure on Localized Quantum Coordinate Rings
• Localizations for quiver Hecke algebras
• Localizations for quiver Hecke algebras III
• Quiver Hecke algebras for Borcherds-Cartan datum II