2016 Fiscal Year Research-status Report
| Project/Area Number |
16K17571
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| Research Institution | Kyoto University |
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Principal Investigator |
IP Ivan 京都大学, スーパーグローバルコース数学系ユニット, 特定助教 (50646031)
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| Project Period (FY) |
2016-04-01 – 2018-03-31
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| Keywords | positive representation / cluster algbera / teichmuller theory / modular double |
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| Outline of Annual Research Achievements |
In FY2016, I followed the research focus I and II outlined in the proposal, namely to study the braided tensor structure of positive representations, as well as the applications to higher Teichmuller theory and its super analogue.
In research focus I, concerning braided tensor category structure, I found a new cluster algebraic realization of quantum groups and its modular double via the positive representations. This is an important result which generalize the recent work of G. Schrader and A. Shapiro. It seems that this cluster algebraic language allows us to study the positive Casimirs, by transforming via cluster mutations to the quantum Hamiltonian operators in certain Today integrable systems, which can be simultaneously diagonalized. In particular this finally proves the long standing conjecture of closure under taking tensor product of the positive representations in type An.
In research focus II, together with A. Zeitlin and R. Penner, we successfully constructed the N=2 super decorated Teichmuller theory, which is important in superstring theory with structure group Osp(2|2) and generalize the previous work on N=1 case. We associated to each triangles of a punctured triangulated surface a new set of coordinates defined by 5 even variables and 2 odd variables, and give a new explicitly description of the spin structures using the language of G-graph connection.
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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
We have published one preprint on N=2 super Teichmuller theory, and one 90-page preprint on cluster realization of quantum groups. Both are submitted to journals for reviews. The results have been presented in various seminars and conferences. Currently we are getting more researchers interested in the theory of positive representations, and we expect more collaborations in the future. In particular, I have met with G. Schrader and A. Shapiro, and together we have a clear idea of finally proving the long standing conjecture of braided tensor category structure in type An, as well as the Peter-Weyl theorem, and various other structural properties using the cluster realization of quantum groups found in this project in Fy2016.
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| Strategy for Future Research Activity |
We will invite G. Schrader (Columbia University), A. Shapiro (University of Toronto) as well as I. Le (Perimeter Institute) to further discuss the applications of the cluster realization of quantum groups to the structural properties of positive representations, as well as its relation to higher Teichmuller theory and root of unity limit.
I also plan to invite or visit again A. Zeitlin (Louisiana State University) to further study our project on super higher Teichmuller theory, concerning the dimension reduction of the Ramond punctures in the super Riemannian manifold, thus refining our previous results of the decorated Teichmuller 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 invitation for A. Zeitlin from Columbia University was originally planned for FY2016, but it was delayed due to changes in his personal schedule. It will be rescheduled in FY2017. The planned visit to Toronto is scheduled on April 2017, hence the part of grant for the visit is carried over to FY2017.
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| Expenditure Plan for Carryover Budget |
Invitation of collaborators A. Zeitlin (USA), G. Schrader (USA), A. Shapiro (Canada) and I. Le (Canada) will be scheduled on the fall of 2017 and spring of 2018, each with around 200,000JPY budget.
Attending international conferences (transport and local expenses) 400,000 JPY, local workshops (transport and local expenses) 200,000JPY, and purchases of books for research purpose: 100,000JPY.
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Research Products
(8 results)
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[Presentation] Split Real Quantum Groups2016
Author(s)
Ivan Chi Ho Ip
Organizer
The 24th International Conference on Integrable Systems and Quantum symmetries
Place of Presentation
CVUT, Prague, Czech Republic
Year and Date
2016-06-14 – 2016-06-18
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