2017 Fiscal Year Annual Research 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 algebra / quantum groups / Peter Weyl / Ramond puncture / Teichmuller theory |
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| Outline of Annual Research Achievements |
In FY 2017, we continue the study of properties of positive representations and its applications.
In research focus I, as proposed I have invited Gus Schrader (Columbia University), Alexander Shapiro (University of Toronto) and Ian Le (Perimeter Institute) to Kyoto University to further discuss on the Peter-Weyl theorem of split real quantum group, proving a long standing conjecture in the case of type An. This utilizes the cluster realization of positive representations discovered in FY2016, and also a generalization of the C*-algebraic aspects of split real quantum groups constructed in my previous publications. The results will be published as soon as possible after finalizing more details and depend on the progress of my collaborators.
Further projects with Ian Le also include the braiding action on the cluster realization associated to a triangle, which provide us an S3 action on the cluster structure and allow us to write the universal R operator in a canonical way, answering a conjecture proposed in my previous publication of cluster realization of quantum groups in FY2016.
In research focus II, together with Anton Zeitlin (Louisiana State University) and Robert Penner (IHES), we continue our study of N=2 super Teichmuller theory. In FY2017, we studied 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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