Quantum Cluster Algebra Structures on Quantum Nilpotent Algebras.

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

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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.