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Quantum Cluster Algebra Structures on Quantum Nilpotent Algebras
 
K. R. Goodearl University of California, Santa Barbara
M. T. Yakimov Louisiana State University, Baton Rouge
Quantum Cluster Algebra Structures on Quantum Nilpotent Algebras
eBook ISBN:  978-1-4704-3699-5
Product Code:  MEMO/247/1169.E
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $45.00
Quantum Cluster Algebra Structures on Quantum Nilpotent Algebras
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Quantum Cluster Algebra Structures on Quantum Nilpotent Algebras
K. R. Goodearl University of California, Santa Barbara
M. T. Yakimov Louisiana State University, Baton Rouge
eBook ISBN:  978-1-4704-3699-5
Product Code:  MEMO/247/1169.E
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $45.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2472016; 119 pp
    MSC: Primary 16; Secondary 13; 17; 14

    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

  • Table of Contents
     
     
    • Chapters
    • 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
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2472016; 119 pp
MSC: Primary 16; Secondary 13; 17; 14

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

  • Chapters
  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.