2014 Fiscal Year Research-status Report
| Project/Area Number |
26800004
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| Research Institution | The University of Tokyo |
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Principal Investigator |
IP Ivan 東京大学, カブリ数物連携宇宙研究機構, 特任研究員 (50646031)
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| Project Period (FY) |
2014-04-01 – 2016-03-31
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| Keywords | Positive Representation / Modular Double / Teichmuller Theory / Casimir operators / TQFT / cluster algebra / quantum Lie superalgebra |
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| Outline of Annual Research Achievements |
In FT2014, I followed the Research Focus I and II outline in the application. Namely we continue to study the positive representation of split real quantum groups and its braided category structure, as well as its application to higher quantum Teichmuller theory.
Considering Research Focus I ― Braided Tensor Category and Application, the steps toward analyzing the braiding structure is to establish the action of the Casimir operators and their eigenvalue decomposition. During the project, I have calculated the central characters of the Casimir operators acting on the positive representations, and found a new relation to the theory of primitive forms and discriminant varieties related to other algebraic geometry problems. Together with M. Yamazaki (IPMU), we have also established the behavior of the quantum dilogarithms restricted to the root of unity case and studied new results about the generalized pentagon relations. This result on the special functions will be important to the theory of positive representation since they are the building blocks of the matrix coefficients of the representations.
Concerning Research Focus II ― Higher Teichmuller Theory and Positive Representation, collaboration was done with H. K. Kim (KIAS) to establish the required T and A operators of the theory constructed from the positive representation. However, the A operators thus obtained require further modification in order for it to satisfy the Kashaev’s compatibility equations, and the current research is still on-going. A separate publication is expected to be done in FY2015.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Three preprints on the research project have been published with one of them being accepted by the International Mathematics Research Notices (2015). The research progress is reported in various international conferences during FY2014. I have also attended other workshops and conferences on categorification and cluster algebra theory to enhance communication with other researchers interested in the current research project.
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| Strategy for Future Research Activity |
The invitations of my collaborator A. Zeitlin (Columbia University) on Research Focus III about quantized Lie superalgebra and its relation to positive representations, as well as W. Wong (University of Ottawa) on Research Focus I about braided tensor category related to quantum tori were pushed back to FY2015. I will continue the collaboration on the research project on establishing the A operator with H. K. Kim (KIAS).
Furthermore, as an extension of the project with M. Yamazaki, we will continue exploring the behavior of the quantum dilogarithms restricted to root of unity in the case when the b parameter is pure imaginary, corresponding to the strong regime of the quantized Liouville theory.
Finally, I will plan on further attending international conferences and local workshops to present the results from the current research project.
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| Causes of Carryover |
The invitations of both collaborator A. Zeitlin from Columbia University and W. Wong from University of Ottawa were originally planned for FY2014. However due to changes in their personal schedule, their visit were delayed to after 2015 April. Therefore the part of the grant required for their visit is carried over to FY2015. Furthermore the plans for purchasing books and computer for research purpose has not occurred in FY2014, if grant permits they will be spent in FY2015.
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| Expenditure Plan for Carryover Budget |
Invitation of collaborator A. Zeitlin (USA): 200,000 JPY、Invitation of collaborator W. Wong (Canada): 200,000 JPY、International Conferences (transport and local expense): 400,000 JPY、Local Workshops, (transport and local expense): 200,000 JPY、Purchasing of books and computer for research purpose: 300,000 JPY
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Research Products
(5 results)
