Quantum cluster algebras and quantum nilpotent algebras

Abstract

A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of application of the general results to the above mentioned types of problems. As a consequence, we prove the Berenstein--Zelevinsky conjecture for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß, Leclerc and Schröer for the case of symmetric Kac--Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein, Fomin and Zelevinsky associated to double Bruhat cells coincide with the corresponding cluster algebras.

References in corpus (3)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Quantum cluster algebras and quantum nilpotent algebras
• Cluster algebras in algebraic Lie theory

Cited by in corpus (22)

• Quantum cluster algebras and quantum nilpotent algebras
• Factorizations in bounded hereditary Noetherian prime rings
• Factorizations of Elements in Noncommutative Rings: A Survey
• The Berenstein-Zelevinsky quantum cluster algebra conjecture
• Cluster algebra structures on Poisson nilpotent algebras
• Twist automorphisms on quantum unipotent cells and dual canonical bases
• Factoriality and class groups of cluster algebras
• Bott-Samelson varieties and Poisson Ore extensions
• Prime factors of quantum Schubert cell algebras and clusters for quantum Richardson varieties
• Dual canonical bases and quantum cluster algebras
• Integral quantum cluster structures
• On the T-leaves of some Poisson structures related to products of flag varieties
• Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras
• Graded automorphisms of quantum affine spaces
• Double Bruhat cells and symplectic groupoids
• Cluster algebras of finite type via a Coxeter element and Demazure Crystals of type B,C,D
• Bott-Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces
• On inner Poisson structures of a quantum cluster algebra without coefficients
• Cluster algebras and their bases
• Poisson structure and second quantization of quantum cluster algebras
• Cluster monomials in $\mathbb{C}[G{L}_n/N]$, a simplicial fan in the cone of semi-standard Young tableaux, and the Lusztig basis
• Double Quantum Schubert Cells and Quantum Mutations