On the Degenerate Multiplicity of the sl_2 Loop Algebra for the 6V Transfer Matrix at Roots of Unity
We review the main result of cond-mat/0503564. The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the $sl_2$ loop algebra symmetry if the $q$ parameter is given by a root of unity, $q_0^{2N}=1$, for an integer $N$. We discuss the dimensions of the degenerate ei...
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2006-02-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Online Access: | http://www.emis.de/journals/SIGMA/2006/Paper021/ |
| _version_ | 1819208571229831168 |
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| author | Tetsuo Deguchi |
| author_facet | Tetsuo Deguchi |
| author_sort | Tetsuo Deguchi |
| collection | DOAJ |
| description | We review the main result of cond-mat/0503564. The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the $sl_2$ loop algebra symmetry if the $q$ parameter is given by a root of unity, $q_0^{2N}=1$, for an integer $N$. We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight ${ar d}_k^{pm}$, which leads to evaluation parameters $a_j$. If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state. |
| first_indexed | 2024-12-23T05:41:30Z |
| format | Article |
| id | doaj.art-5a16e51236394a3d8049fc7bc7755cd9 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-23T05:41:30Z |
| publishDate | 2006-02-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-5a16e51236394a3d8049fc7bc7755cd92022-12-21T17:58:11ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592006-02-012021On the Degenerate Multiplicity of the sl_2 Loop Algebra for the 6V Transfer Matrix at Roots of UnityTetsuo DeguchiWe review the main result of cond-mat/0503564. The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the $sl_2$ loop algebra symmetry if the $q$ parameter is given by a root of unity, $q_0^{2N}=1$, for an integer $N$. We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight ${ar d}_k^{pm}$, which leads to evaluation parameters $a_j$. If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state.http://www.emis.de/journals/SIGMA/2006/Paper021/loop algebrathe six-vertex modelroots of unity representations of quantum groupsDrinfeld polynomial |
| spellingShingle | Tetsuo Deguchi On the Degenerate Multiplicity of the sl_2 Loop Algebra for the 6V Transfer Matrix at Roots of Unity Symmetry, Integrability and Geometry: Methods and Applications loop algebra the six-vertex model roots of unity representations of quantum groups Drinfeld polynomial |
| title | On the Degenerate Multiplicity of the sl_2 Loop Algebra for the 6V Transfer Matrix at Roots of Unity |
| title_full | On the Degenerate Multiplicity of the sl_2 Loop Algebra for the 6V Transfer Matrix at Roots of Unity |
| title_fullStr | On the Degenerate Multiplicity of the sl_2 Loop Algebra for the 6V Transfer Matrix at Roots of Unity |
| title_full_unstemmed | On the Degenerate Multiplicity of the sl_2 Loop Algebra for the 6V Transfer Matrix at Roots of Unity |
| title_short | On the Degenerate Multiplicity of the sl_2 Loop Algebra for the 6V Transfer Matrix at Roots of Unity |
| title_sort | on the degenerate multiplicity of the sl 2 loop algebra for the 6v transfer matrix at roots of unity |
| topic | loop algebra the six-vertex model roots of unity representations of quantum groups Drinfeld polynomial |
| url | http://www.emis.de/journals/SIGMA/2006/Paper021/ |
| work_keys_str_mv | AT tetsuodeguchi onthedegeneratemultiplicityofthesl2loopalgebraforthe6vtransfermatrixatrootsofunity |