Spherical Representations of *-Flows II: Representation System and Quantum Group Setup
This paper is a sequel to our previous study of spherical representations in the operator algebra setup. We first introduce possible analogs of dimension groups in the present context by utilizing the notion of operator systems and their relatives. We then apply our study to inductive limits of comp...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211618 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Spherical Representations of *-Flows II: Representation System and Quantum Group Setup. Yoshimichi Ueda. SIGMA 18 (2022), 050, 43 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859715350224961536 |
|---|---|
| author | Ueda, Yoshimichi |
| author_facet | Ueda, Yoshimichi |
| citation_txt | Spherical Representations of *-Flows II: Representation System and Quantum Group Setup. Yoshimichi Ueda. SIGMA 18 (2022), 050, 43 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | This paper is a sequel to our previous study of spherical representations in the operator algebra setup. We first introduce possible analogs of dimension groups in the present context by utilizing the notion of operator systems and their relatives. We then apply our study to inductive limits of compact quantum groups, and establish an analogue of Olshanski's notion of spherical unitary representations of infinite-dimensional Gelfand pairs of the form < × (via the diagonal embedding) in the quantum group setup. This, in particular, justifies Ryosuke Sato's approach to asymptotic representation theory for quantum groups.
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| first_indexed | 2026-03-15T08:19:25Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211618 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T08:19:25Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ueda, Yoshimichi 2026-01-07T13:38:50Z 2022 Spherical Representations of *-Flows II: Representation System and Quantum Group Setup. Yoshimichi Ueda. SIGMA 18 (2022), 050, 43 pages 1815-0659 2020 Mathematics Subject Classification: 22D25; 22E66; 46L67; 17B37 arXiv:2201.10931 https://nasplib.isofts.kiev.ua/handle/123456789/211618 https://doi.org/10.3842/SIGMA.2022.050 This paper is a sequel to our previous study of spherical representations in the operator algebra setup. We first introduce possible analogs of dimension groups in the present context by utilizing the notion of operator systems and their relatives. We then apply our study to inductive limits of compact quantum groups, and establish an analogue of Olshanski's notion of spherical unitary representations of infinite-dimensional Gelfand pairs of the form < × (via the diagonal embedding) in the quantum group setup. This, in particular, justifies Ryosuke Sato's approach to asymptotic representation theory for quantum groups. The author thanks Ryosuke Sato for many discussions on his works on the asymptotic representation theory for quantum groups. The author also thanks Reiji Tomatsu and Yuki Arano for explaining many things about quantum groups and tensor categories during the last decade. In fact, this work benefited from what Reiji and Yuki explained to the author on various occasions. Finally, the author thanks Makoto Yamashita for presenting him with his Japanese book [41] some years ago. The book was so helpful to the author in getting a perspective on quantum groups before this work. Finally, the author thanks the referees for their comments. This work was supported by Grant-in-Aid for Scientific Research (B) JP18H01122. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Spherical Representations of *-Flows II: Representation System and Quantum Group Setup Article published earlier |
| spellingShingle | Spherical Representations of *-Flows II: Representation System and Quantum Group Setup Ueda, Yoshimichi |
| title | Spherical Representations of *-Flows II: Representation System and Quantum Group Setup |
| title_full | Spherical Representations of *-Flows II: Representation System and Quantum Group Setup |
| title_fullStr | Spherical Representations of *-Flows II: Representation System and Quantum Group Setup |
| title_full_unstemmed | Spherical Representations of *-Flows II: Representation System and Quantum Group Setup |
| title_short | Spherical Representations of *-Flows II: Representation System and Quantum Group Setup |
| title_sort | spherical representations of *-flows ii: representation system and quantum group setup |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211618 |
| work_keys_str_mv | AT uedayoshimichi sphericalrepresentationsofflowsiirepresentationsystemandquantumgroupsetup |