| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 30, 2013 |
| Latest Amendment Date: | July 30, 2013 |
| Award Number: | 1303038 |
| Award Instrument: | Standard Grant |
| Program Manager: |
James Matthew Douglass
mdouglas@nsf.gov (703)292-2467 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 15, 2013 |
| End Date: | July 31, 2017 (Estimated) |
| Total Intended Award Amount: | $164,000.00 |
| Total Awarded Amount to Date: | $164,000.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 (225)578-2760 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
202 Himes Hall Baton Rouge LA US 70803-2701 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | ALGEBRA,NUMBER THEORY,AND COM |
| Primary Program Source: |
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| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
The PI will work on a series of interrelated problems for quantum groups and quantum cluster algebras. Firstly, he will work on proving that each member of a very general axiomatically defined class of quantum nilpotent algebras admits a canonical quantum cluster algebra structure. Based on this, he will attempt to construct a unified categorification for all algebras in the class. Many important families arise as special cases of algebras in this large class, most notably the quantum double Bruhat cell algebras and the quantum Schubert cell algebras. The former case will lead to a proof of the Berenstein-Zelevinsky quantum cluster algebra conjecture. In both cases the quantum cluster algebra structure will be used to study the topology of the spectra of quantum groups and quantum Schubert cell algebras. In the former case he will attempt to prove bicontinuity of his recently constructed Dixmier map from the symplectic foliation on a simple Lie group equipped with the so called standard Poisson structure to the primitive spectrum of a quantum group. The PI and his graduate students will apply these ideas for proving the existence of quantum foldings and for building quantum cluster algebras from them. He will also apply his recent results on rigidity of quantum tori to the classification of automorphism groups of interesting (quantum) cluster algebras. Via the work of Gekhtman, Shapiro and Vainshtein a certain large class of classical cluster algebras can be approached using Poisson geometry. The PI will work on Poisson analogs of the above projects using a notion of Poisson unique factorization domains.
Noncommutative and Poisson algebras arise in many different aspects of mathematics (functions on geometric objects) and physics (observables in classical and quantum mechanical systems). These objects are studied using many different techniques on the basis of algebraic, geometric, analytic and combinatorial methods. The PI will study these objects via two different methods. The first one is a classical one, based on studying the presentations of these algebras as collections of operators (representations). This method uses techniques from algebra and geometry. The second method is based on the recent combinatorial notion of cluster algebras invented by Fomin and Zelevinsky. It leads to a very concrete combinatorial structure on the objects. The idea of mutation is then used to study various parts of the objects which are not seen by the previous methods (they focused on a particular "initial side" of these algebras). Using his recent rigidity results, the PI will also study and classify the symmetries of the objects in the above classes. The motivation for this is that symmetries reduce the complexity of an abject and the full description of the collection of symmetries provides an understanding of the complexity of the object.
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
Cluster Algebras were defined by S. Fomin and A. Zelevinsky in groundbreaking works from 2002. Within very short time they started playing a major role in many areas in Mathematics and Mathematical Physics. One of the key open problems in the area was to prove that quantized coordinate rings of important varieties in representation theory have structures of quantum cluster algebras. This is a major step towards understanding the intrinsic structure of their canonical bases in terms of cluster algebras. Goodearl and the PI settled several key open problems in this respect, most notably the Berenstein-Zelevinsky conjecture that the coordinate rings of double Bruhat cells posses canonical strictures of cluster algebras. In a subsequent work, Lenagan and the PI treated the problem of constructing sufficiently many clusters for the coordinate rings of Richardson varieties.
In a related direction, Levitt and the PI proved a rigidity theorem that can be used to classify the full sets of symmetries of Cluster Algebras. This method has a broad range of important applications because for a many reasons we need to know the full sets of symmetries of geometric and algebraic objects, and here the results describe these symmetries for the important class of Cluster Algebras. Bucher and the PI obtained explicit results for the family of the cluster algebras of oriented surfaces of Fomin, Shapiro and Thurston. These algebras play a central role in the area due to their multiple relations to problems in geometry, combinatorics and mathematical physics.
Calabi-Yau categories are fundamental objects in geometry, representation theory and mathematical physics. Jorgensen and the PI constructed invariants for such categories which attach a Kac-Moody group to each of them. This work brings representation theoretic tools to the study of the structure of such categories, and, in the opposite direction, categorifies the root systems of simple Lie groups in terms of Calabi-Yau categories. The latter, in turn, can be used to study the representations of these groups by the abstract methods of triangulated categories.
The prime spectra of noncommutative algebras are major geometric objects that one attaches to noncommutative algebras. The PI, jointly with Fryer and Lenagan, obtained a number of results on the prime factors of quantum groups. These concern their separation properties and the existence of quantum clusters. These results provide key steps towards the solution of the old problem of describing the topology of the spectra of quantum groups.
Discriminants play a fundamental role in various settings in algebraic number theory, algebraic geometry, combinatorics, and noncommutative algebra. In the last case, they have been computed for very few algebras. The PI and his graduate students Nguyen and Trampel developed a general method for computing discriminants of noncommutative algebras which is applicable to algebras obtained by specialization from families, such as quantum algebras at roots of unity. It built a connection with Poisson geometry and expressed the discriminants as products of Poisson primes. From a different perspective it related noncommutative discriminants to frozen variables of quantum cluster algebras. This method settled many open problems in the area on the computation of discriminants of algebras that appear in the theory of quantum groups.
The bispectral problem, of Duistermaat and Grunbaum is an old problem in classical analysis about classifying complex analytic functions in two variablescthat are eigenfunctions in each of them. The PI, together with his former PhD student Geiger and with Horozov, proved a general theorem that establishes the bispectrality of noncommutative Darboux transformations in very broad settings. This method links the very distant areas of classical analysis and noncommutative algebra. With the help of the latter, the method produces vast classes of bispectral functions that were not treatable with any other methods before.
Last Modified: 10/30/2017
Modified by: Milen T Yakimov
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