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Quantum cluster algebra structures on quantum nilpotent algebras

About this Title

K. R. Goodearl, Department of Mathematics, University of California, Santa Barbara, California 93106 and M. T. Yakimov, Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 247, Number 1169
ISBNs: 978-1-4704-3694-0 (print); 978-1-4704-3699-5 (online)
Published electronically: December 28, 2016
Keywords: Quantum cluster algebras, quantum nilpotent algebras, iterated Ore extensions, noncommutative unique factorization domains
MSC: Primary 16T20; Secondary 13F60, 17B37, 14M15

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Table of Contents


  • 1. Introduction
  • 2. Quantum cluster algebras
  • 3. Iterated skew polynomial algebras and noncommutative UFDs
  • 4. One-step mutations in CGL extensions
  • 5. Homogeneous prime elements for subalgebras of symmetric CGL extensions
  • 6. Chains of mutations in symmetric CGL extensions
  • 7. Division properties of mutations between CGL extension presentations
  • 8. Symmetric CGL extensions and quantum cluster algebras
  • 9. Quantum groups and quantum Schubert cell algebras
  • 10. Quantum cluster algebra structures on quantum Schubert cell algebras


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.

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