Monoidal categorification of cluster algebras II

Abstract

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 $q^{1/2}$.

arXiv comment51pages,

References in corpus (9)

• A diagrammatic approach to categorification of quantum groups I
• 2-Kac-Moody algebras
• Kac-Moody groups and cluster algebras
• Quiver varieties and cluster algebras
• Semicanonical bases and preprojective algebras
• Graded quiver varieties, quantum cluster algebras and dual canonical basis
• Cluster algebra structures and semicanonical bases for unipotent groups
• Purity for graded potentials and quantum cluster positivity
• Positivity for cluster algebras

Cited by in corpus (14)

• Triangular bases in quantum cluster algebras and monoidal categorification conjectures
• Geometric conditions for $\square$-irreducibility of certain representations of the general linear group over a non-archimedean local field
• The Berenstein-Zelevinsky quantum cluster algebra conjecture
• Advances in R-matrices and their applications (after Maulik-Okounkov, Kang-Kashiwara-Kim-Oh,...)
• Dual canonical bases and quantum cluster algebras
• Folding KLR algebras
• Affinizations and R-matrices for quiver Hecke algebras
• On Structure of cluster algebras of geometric type, II: Green's equivalences and paunched surfaces
• From frieze patterns to cluster categories
• Categorical relations between Langlands dual quantum affine algebras: Doubly laced types
• Monoidal categories of modules over quantum affine algebras of type A and B
• Symmetrization of representations of $G{L}_N$
• Convolution Products and R-Matrices for KLR Algebras of Type $B$
• Cluster algebras and their bases