2019/11/29 by Léa Bittmann, Bittmann, Léa · 3 citations
Mathematics · #16T20 17B37 13F60 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1911.13110
openalex publication_date 2019/11/29 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We establish a quantum cluster algebra structure on the quantum Grothendieck\nring of a certain monoidal subcategory of the category of finite-dimensional\nrepresentations of a simply-laced quantum affine algebra. Moreover, the\n(q,t)-characters of certain irreducible representations, among which\nfundamental representations, are obtained as quantum cluster variables. This\napproach gives a new algorithm to compute these (q,t)-characters. As an\napplication, we prove that the quantum Grothendieck ring of a larger category\nof representations of the Borel subalgebra of the quantum affine algebra,\ndefined in a previous work as a quantum cluster algebra, contains indeed the\nwell-known quantum Grothendieck ring of the category of finite-dimensional\nrepresentations. Finally, we display our algorithm on a concrete example.\n