A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions
For a finite-dimensional simple Lie algebra g, let U⁺q(g) be the positive part of the quantized universal enveloping algebra, and Aq(g) be the quantized algebra of functions. We show that the transition matrix of the PBW bases of U⁺q(g) coincides with the intertwiner between the irreducible Aq(g)-mo...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2013 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2013
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| Zitieren: | A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Function / A. Kuniba, M. Okado, Y. Yamada // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 27 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862664224747028480 |
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| author | Kuniba, A. Okado, M. Yamada, Y. |
| author_facet | Kuniba, A. Okado, M. Yamada, Y. |
| citation_txt | A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Function / A. Kuniba, M. Okado, Y. Yamada // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 27 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | For a finite-dimensional simple Lie algebra g, let U⁺q(g) be the positive part of the quantized universal enveloping algebra, and Aq(g) be the quantized algebra of functions. We show that the transition matrix of the PBW bases of U⁺q(g) coincides with the intertwiner between the irreducible Aq(g)-modules labeled by two different reduced expressions of the longest element of the Weyl group of g. This generalizes the earlier result by Sergeev on A₂ related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for C₂. Our proof is based on a realization of U⁺q(g) in a quotient ring of Aq(g).
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| first_indexed | 2025-12-07T15:13:52Z |
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| id | nasplib_isofts_kiev_ua-123456789-149342 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:13:52Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
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| spelling | Kuniba, A. Okado, M. Yamada, Y. 2019-02-21T07:04:16Z 2019-02-21T07:04:16Z 2013 A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Function / A. Kuniba, M. Okado, Y. Yamada // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 20G42; 81R50; 17B80 DOI: http://dx.doi.org/10.3842/SIGMA.2013.049 https://nasplib.isofts.kiev.ua/handle/123456789/149342 For a finite-dimensional simple Lie algebra g, let U⁺q(g) be the positive part of the quantized universal enveloping algebra, and Aq(g) be the quantized algebra of functions. We show that the transition matrix of the PBW bases of U⁺q(g) coincides with the intertwiner between the irreducible Aq(g)-modules labeled by two different reduced expressions of the longest element of the Weyl group of g. This generalizes the earlier result by Sergeev on A₂ related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for C₂. Our proof is based on a realization of U⁺q(g) in a quotient ring of Aq(g). This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full
 collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html.
 The authors thank Ivan C.H. Ip, Anatol N. Kirillov, Toshiki Nakashima and Masatoshi Noumi
 for communications. They also thank one of the referees for drawing attention to the references [9, 26]. This work is supported by Grants-in-Aid for Scientific Research No. 23340007,
 No. 24540203, No. 23654007 and No. 21340036 from JSPS. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions Article published earlier |
| spellingShingle | A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions Kuniba, A. Okado, M. Yamada, Y. |
| title | A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions |
| title_full | A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions |
| title_fullStr | A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions |
| title_full_unstemmed | A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions |
| title_short | A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions |
| title_sort | common structure in pbw bases of the nilpotent subalgebra of uq(g) and quantized algebra of functions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149342 |
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