A q-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine Algebra
We prove an analogue of the Sylvester theorem for the generator matrices of the quantum affine algebra ${ m U}_q(widehat{mathfrak{gl}}_n)$. We then use it to give an explicit realization of the skew representations of the quantum affine algebra. This allows one to identify them in a simple way by ca...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
National Academy of Science of Ukraine
2006-12-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Subjects: | |
| Online Access: | http://www.emis.de/journals/SIGMA/2006/Paper092/ |
| _version_ | 1818923768388517888 |
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| author | Mark J. Hopkins Alexander I. Molev |
| author_facet | Mark J. Hopkins Alexander I. Molev |
| author_sort | Mark J. Hopkins |
| collection | DOAJ |
| description | We prove an analogue of the Sylvester theorem for the generator matrices of the quantum affine algebra ${ m U}_q(widehat{mathfrak{gl}}_n)$. We then use it to give an explicit realization of the skew representations of the quantum affine algebra. This allows one to identify them in a simple way by calculating their highest weight, Drinfeld polynomials and the Gelfand-Tsetlin character (or $q$-character). We also apply the quantum Sylvester theorem to construct a$q$-analogue of the Olshanski algebra as a projective limit of certaincentralizers in ${ m U}_q(mathfrak{gl}_n)$ and show that this limit algebra contains the $q$-Yangian as a subalgebra. |
| first_indexed | 2024-12-20T02:14:41Z |
| format | Article |
| id | doaj.art-219f891f020349a79b45627affb83f1a |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-20T02:14:41Z |
| publishDate | 2006-12-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-219f891f020349a79b45627affb83f1a2022-12-21T19:56:58ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592006-12-012092A q-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine AlgebraMark J. HopkinsAlexander I. MolevWe prove an analogue of the Sylvester theorem for the generator matrices of the quantum affine algebra ${ m U}_q(widehat{mathfrak{gl}}_n)$. We then use it to give an explicit realization of the skew representations of the quantum affine algebra. This allows one to identify them in a simple way by calculating their highest weight, Drinfeld polynomials and the Gelfand-Tsetlin character (or $q$-character). We also apply the quantum Sylvester theorem to construct a$q$-analogue of the Olshanski algebra as a projective limit of certaincentralizers in ${ m U}_q(mathfrak{gl}_n)$ and show that this limit algebra contains the $q$-Yangian as a subalgebra.http://www.emis.de/journals/SIGMA/2006/Paper092/quantum affine algebraquantum Sylvester theoremskew representations |
| spellingShingle | Mark J. Hopkins Alexander I. Molev A q-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine Algebra Symmetry, Integrability and Geometry: Methods and Applications quantum affine algebra quantum Sylvester theorem skew representations |
| title | A q-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine Algebra |
| title_full | A q-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine Algebra |
| title_fullStr | A q-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine Algebra |
| title_full_unstemmed | A q-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine Algebra |
| title_short | A q-Analogue of the Centralizer Construction and Skew Representations of the Quantum Affine Algebra |
| title_sort | q analogue of the centralizer construction and skew representations of the quantum affine algebra |
| topic | quantum affine algebra quantum Sylvester theorem skew representations |
| url | http://www.emis.de/journals/SIGMA/2006/Paper092/ |
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