Monoidal categorification of cluster algebras II Article Swipe

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Categorification Cluster algebra Mathematics Pure mathematics Conjecture Algebra over a field Basis (linear algebra) Quantum Hopf algebra Unipotent Physics Geometry Quantum mechanics

Seok‐Jin Kang , Masaki Kashiwara , Myungho Kim , Se‐jin Oh ·

YOU? · · 2015 · Open Access · · DOI: https://doi.org/10.48550/arxiv.1502.06714 · OA: W2149963522

We prove that the quantum unipotent coordinate algebra $A_q(\mathfrak{n}(w))\ $ associated with a symmetric Kac-Moody algebra and its Weyl group element $w$ has a monoidal categorification as a quantum cluster algebra. As an application of our earlier work, we achieve it by showing the existence of a quantum monoidal seed of $A_q(\mathfrak{n}(w))$ which admits the first-step mutations in all the directions. As a consequence, we solve the conjecture that any cluster monomial is a member of the upper global basis up to a power of $q^{1/2}$.