2017 Fiscal Year Final Research Report
| Project/Area Number |
16K17571
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Algebra
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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 / quantum groups / modular double / Teichmuller theory / cluster algebra / integrable systems |
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| Outline of Final Research Achievements |
We discovered a cluster realization of the positive representations of split real quantum groups and factorization of universal R operator. With this new result, we found explicitly the tensor product decomposition of positive representations restricted to the Borel part. The same technique is further developed by Schrader-Shapiro to prove that the positive representations in type An is closed under taking tensor product by identifying the Casimirs with the open Coxeter-Toda Hamiltonians, proving part of the main conjecture started for this project. In an upcoming work, we also show that the Peter-Weyl theorem holds using the same decomposition.
With R. Penner and A. Zeitlin, we also constructed the N=2 super decorated Teichmuller theory, and described the dimension reduction of the odd coordinates of the super Teichmuller space due to the constraint arising from Ramond punctures on the surface, so that the result is compatible with that of the moduli spaces of super Riemann surfaces.
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| Free Research Field |
Representation Theory
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