Quantum cluster algebra structures on quantum nilpotent algebras

Abstract

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to these problems, including the construction of quantum cluster algebra structures on quantum unipotent groups and quantum double Bruhat cells (the Berenstein-Zelevinsky conjecture), and treat these problems from a unified perspective. All such applications also establish equality between the constructed quantum cluster algebras and their upper counterparts. The proofs rely on Chatters' notion of noncommutative unique factorization domains. Toric frames are constructed by considering sequences of homogeneous prime elements of chains of noncommutative UFDs (a generalization of the construction of Gelfand-Tsetlin subalgebras) and mutations are obtained by altering chains of noncommutative UFDs. Along the way, an intricate (and unified) combinatorial model for the homogeneous prime elements in chains of noncommutative UFDs and their alterations is developed. When applied to special families, this recovers the combinatorics of Weyl groups and double Weyl groups previously used in the construction and categorification of cluster algebras. It is expected that this combinatorial model of sequences of homogeneous prime elements will have applications to the unified categorification of quantum nilpotent algebras.

Final version; accepted for Memoirs Amer. Math. Soc. Minor corrections; running example added; references updated; index added (requires MakeIndex and additional LaTeX runs)

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• Cluster structures on quantum coordinate rings
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• Quantum cluster algebra structures on quantum nilpotent algebras
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Cited by in corpus (24)

• Quantum cluster algebra structures on quantum nilpotent algebras
• 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
• From quantum Ore extensions to quantum tori via noncommutative UFDs
• Twist automorphisms on quantum unipotent cells and dual canonical bases
• Cluster algebra structures on Poisson nilpotent algebras
• 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
• On the T-leaves of some Poisson structures related to products of flag varieties
• Integral quantum cluster structures
• Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras
• Graded automorphisms of quantum affine spaces
• Double Bruhat cells and symplectic groupoids
• Double Quantum Schubert Cells and Quantum Mutations
• Cluster algebras and their bases
• Poisson structure and second quantization of quantum cluster algebras
• Bott-Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces
• On inner Poisson structures of a quantum cluster algebra without coefficients
• Cluster monomials in $\mathbb{C}[G{L}_n/N]$, a simplicial fan in the cone of semi-standard Young tableaux, and the Lusztig basis
• Cluster algebras of finite type via a Coxeter element and Demazure Crystals of type B,C,D