Braid group action on the module category of quantum affine algebras

2020/04/10 by Masaki Kashiwara, Myungho Kim, Kashiwara, Masaki +5 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2004.04939

Abstract

Let \mathfrakg₀ be a simple Lie algebra of type ADE and let U'_q(\mathfrakg) be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group B(\mathfrakg₀) on the quantum Grothendieck ring K_t(\mathfrakg) of Hernandez-Leclerc's category C_\mathfrakg⁰. Focused on the case of type A_N-1, we construct a family of monoidal autofunctors \\mathscrS_i\_{i∈ ℤ} on a localization T_N of the category of finite-dimensional graded modules over the quiver Hecke algebra of type A_∞. Under an isomorphism between the Grothendieck ring K(T_N) of T_N and the quantum Grothendieck ring K_t(A⁽¹⁾_N-1), the functors \\mathscrS_i\_{1≤ i≤ N-1} recover the action of the braid group B(A_N-1). We investigate further properties of these functors.

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• Faithful action of braid group on bosonic extensions

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