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March 2021 Braid group action on the module category of quantum affine algebras
Masaki Kashiwara, Myungho Kim, Se-jin Oh, Euiyong Park
Proc. Japan Acad. Ser. A Math. Sci. 97(3): 13-18 (March 2021). DOI: 10.3792/pjaa.97.003

Abstract

Let $\mathfrak{g}_{0}$ be a simple Lie algebra of type ADE and let $U'_{q}(\mathfrak{g})$ be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group $B(\mathfrak{g}_{0})$ on the quantum Grothendieck ring $\mathcal{K}_{t}(\mathfrak{g})$ of Hernandez-Leclerc’s category $\mathcal{C}_{\mathfrak{g}}^{0}$. Focused on the case of type $A_{N-1}$, we construct a family of monoidal autofunctors $\{\mathcal{S}_{i}\}_{i\in \mathbf{Z}}$ on a localization $\mathcal{T}_{N}$ of the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty}$. Under an isomorphism between the Grothendieck ring $K(\mathcal{T}_{N})$ of $\mathcal{T}_{N}$ and the quantum Grothendieck ring $\mathcal{K}_{t}(A^{(1)}_{N-1})$, the functors $\{\mathcal{S}_{i}\}_{1\leq i\leq N-1}$ recover the action of the braid group $B(A_{N-1})$. We investigate further properties of these functors.

Citation

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Masaki Kashiwara. Myungho Kim. Se-jin Oh. Euiyong Park. "Braid group action on the module category of quantum affine algebras." Proc. Japan Acad. Ser. A Math. Sci. 97 (3) 13 - 18, March 2021. https://doi.org/10.3792/pjaa.97.003

Information

Published: March 2021
First available in Project Euclid: 17 March 2021

Digital Object Identifier: 10.3792/pjaa.97.003

Subjects:
Primary: 17B37 , 20F36
Secondary: 18D10

Keywords: braid group action , quantum affine algebra , quantum Grothendieck ring , quiver Hecke algebra , R-matrix

Rights: Copyright © 2021 The Japan Academy

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Vol.97 • No. 3 • March 2021
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