Quantum Integrable Model of an Arrangement of Hyperplanes
The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this paper a quantum integrable model is assigned to a weighted arrangement of affine hyperplanes. We show (under certain assumpt...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2011 |
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| Sprache: | English |
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Інститут математики НАН України
2011
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| Zitieren: | Quantum Integrable Model of an Arrangement of Hyperplanes / A. Varchenko // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 29 назв. — англ. |
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Varchenko, A. 2019-02-11T15:45:50Z 2019-02-11T15:45:50Z 2011 Quantum Integrable Model of an Arrangement of Hyperplanes / A. Varchenko // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 82B23; 32S22; 17B81; 81R12 DOI:10.3842/SIGMA.2011.032 https://nasplib.isofts.kiev.ua/handle/123456789/146807 The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this paper a quantum integrable model is assigned to a weighted arrangement of affine hyperplanes. We show (under certain assumptions) that the algebra of Hamiltonians of the model is isomorphic to the algebra of functions on the critical set of the corresponding master function. For a discriminantal arrangement we show (under certain assumptions) that the symmetric part of the algebra of Hamiltonians is isomorphic to the Bethe algebra of the corresponding Gaudin model. It is expected that this correspondence holds in general (without the assumptions). As a byproduct of constructions we show that in a Gaudin model (associated to an arbitrary simple Lie algebra), the Bethe vector, corresponding to an isolated critical point of the master function, is nonzero. This paper is a contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”. The full collection is available at http://www.emis.de/journals/SIGMA/OPSF.html. The idea that an analog of the Bethe ansatz construction does exist for an arbitrary arrangement of hyperplanes was formulated long time ago in [26]. That program had been realized partially in [27]. This paper is an extended exposition of my lectures at Mathematical Society of Japan Seasonal Institute on Arrangements of Hyperplanes in August of 2009. I thank organizers for invitation and Hokkaido University for hospitality. I thank for hospitality Universit´e Paul Sabatier in Toulouse, where this paper had been finished. I thank E. Mukhin, V. Schechtman, V. Tarasov, H. Terao for discussions. The author was supported in part by NSF grant DMS-0555327 en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Quantum Integrable Model of an Arrangement of Hyperplanes Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Quantum Integrable Model of an Arrangement of Hyperplanes |
| spellingShingle |
Quantum Integrable Model of an Arrangement of Hyperplanes Varchenko, A. |
| title_short |
Quantum Integrable Model of an Arrangement of Hyperplanes |
| title_full |
Quantum Integrable Model of an Arrangement of Hyperplanes |
| title_fullStr |
Quantum Integrable Model of an Arrangement of Hyperplanes |
| title_full_unstemmed |
Quantum Integrable Model of an Arrangement of Hyperplanes |
| title_sort |
quantum integrable model of an arrangement of hyperplanes |
| author |
Varchenko, A. |
| author_facet |
Varchenko, A. |
| publishDate |
2011 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this paper a quantum integrable model is assigned to a weighted arrangement of affine hyperplanes. We show (under certain assumptions) that the algebra of Hamiltonians of the model is isomorphic to the algebra of functions on the critical set of the corresponding master function. For a discriminantal arrangement we show (under certain assumptions) that the symmetric part of the algebra of Hamiltonians is isomorphic to the Bethe algebra of the corresponding Gaudin model. It is expected that this correspondence holds in general (without the assumptions). As a byproduct of constructions we show that in a Gaudin model (associated to an arbitrary simple Lie algebra), the Bethe vector, corresponding to an isolated critical point of the master function, is nonzero.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146807 |
| citation_txt |
Quantum Integrable Model of an Arrangement of Hyperplanes / A. Varchenko // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 29 назв. — англ. |
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