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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Höhe: 254 mm
Breite: 178 mm
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ISBN-13
978-1-4704-3694-0 (9781470436940)
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Schweitzer Klassifikation
K. R. Goodearl, University of California, Santa Barbara.
M. T. Yakimov, Louisiana State University, Baton Rouge.
Introduction
Quantum cluster algebras
Iterated skew polynomial algebras and noncommutative UFDs
One-step mutations in CGL extensions
Homogeneous prime elements for subalgebras of symmetric CGL extensions
Chains of mutations in symmetric CGL extensions
Division properties of mutations between CGL extension presentations
Symmetric CGL extensions and quantum cluster algebras
Quantum groups and quantum Schubert cell algebras
Quantum cluster algebra structures on quantum Schubert cell algebras
Bibliography
Index
