A quantum cluster algebra approach to representations of simply-laced quantum affine algebras
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Publication:6330100
DOI10.1007/S00209-020-02664-9zbMATH Open1517.17012arXiv1911.13110MaRDI QIDQ6330100
Publication date: 29 November 2019
Abstract: We establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the (q,t)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these (q,t)-characters. As an application, we prove that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defined in a previous work as a quantum cluster algebra, contains indeed the well-known quantum Grothendieck ring of the category of finite-dimensional representations. Finally, we display our algorithm on a concrete example.
Quantum groups (quantized enveloping algebras) and related deformations (17B37) Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras (17B67) Representations of quivers and partially ordered sets (16G20) Cluster algebras (13F60) Monoidal categories, symmetric monoidal categories (18M05)
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