Cluster structures on quantum coordinate rings

Abstract

We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac-Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory C_w of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities which can be viewed as a q-analogue of a T-system. In case G is a simple algebraic group of type A, D, E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure.

v2: minor corrections. v3: references updated, final version to appear in Selecta Mathematica

Cited by in corpus (54)

• Weak Separation and Plabic Graphs
• Quantum Grothendieck rings and derived Hall algebras
• Graded quiver varieties, quantum cluster algebras and dual canonical basis
• Triangular bases in quantum cluster algebras and monoidal categorification conjectures
• Quantum cluster algebra structures on quantum nilpotent algebras
• Quantum cluster algebras and quantum nilpotent algebras
• On the Spectra of Quantum Groups
• Monoidal categorification and quantum affine algebras
• Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras II
• The Berenstein-Zelevinsky quantum cluster algebra conjecture
• Graded quantum cluster algebras and an application to quantum Grassmannians
• Disordered skyrmion phase stabilized by magnetic frustration in a chiral magnet
• From quantum Ore extensions to quantum tori via noncommutative UFDs
• Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
• Quantum cluster algebras and their specializations
• Isomorphisms among quantum Grothendieck rings and propagation of positivity
• Quantum groups via cyclic quiver varieties I
• A Common Structure in PBW Bases of the Nilpotent Subalgebra of $U_q(\mathfrak{g})$ and Quantized Algebra of Functions
• Twist automorphisms on quantum unipotent cells and dual canonical bases
• Quivers with relations for symmetrizable Cartan matrices V. Caldero-Chapoton formula
• Multi-brace cotensor Hopf algebras and quantum groups
• Dual canonical bases and quantum cluster algebras
• Quantum twist maps and dual canonical bases
• The $(q,t)$-Cartan matrix specialized at $q=1$
• Integral quantum cluster structures
• Dominance order and monoidal categorification of cluster algebras
• Rigidity of quantum tori and the Andruskiewitsch-Dumas conjecture
• An Analog of Leclerc's Conjecture for Bases of Quantum Cluster Algebras
• Root of unity quantum cluster algebras and discriminants
• Unipotent quantum coordinate ring and prefundamental representations for types $A_n^{(1)}$ and $D_n^{(1)}$
• Coherent IC-sheaves on type $A_n$ affine Grassmannians and dual canonical basis of affine type $A_1$
• Triangular bases in quantum cluster algebras
• A cluster realization of $U_q(\mathfrak{s}{\mathfrak{l}}_{\mathfrak{n}})$ from quantum character varieties
• Cluster algebras generated by projective cluster variables
• Quantum deformation of planar amplitudes
• Matrix product solutions to the $G_2$ reflection equation
• Equivariant multiplicities via representations of quantum affine algebras
• Newton-Okounkov polytopes of Schubert varieties arising from cluster structures
• Quantum affine algebras and Grassmannians
• An expansion formula for type A and Kronecker quantum cluster algebras
• The center of ${\mathcal{U}}_q({\mathfrak{n}}_{\omega })$
• Quantisation Spaces of Cluster Algebras
• On homomorphisms from Ringel-Hall algebras to quantum cluster algebras
• Double Quantum Schubert Cells and Quantum Mutations
• Quantized Matrix Algebras and Quantum seeds
• Cluster algebras and their bases
• Twist automorphisms and Poisson structures
• Quantized nilradicals of parabolic subalgebras of $\mathfrak{sl}(n)$ and algebras of coinvariants
• On inner Poisson structures of a quantum cluster algebra without coefficients
• Quantum frieze patterns in quantum cluster algebras of type A
• Quantum Affine Schubert Cells and FRT-Bialgebras: The E_6^{(1)} Case
• Cluster Algebras, Symplectic Leaves and Quantum Groups
• Poisson Ideals in Cluster Algebras and the Spectra of Quantized Coordinate Rings
• Poisson and quantum geometry of acyclic cluster algebras