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Quantum Integrable Model of an Arrangement of Hyperplanes
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2011, том 7
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Quantum Integrable Model of an Arrangement of Hyperplanes
Інший ID:
2010 Mathematics Subject Classification: 82B23; 32S22; 17B81; 81R12
DOI:10.3842/SIGMA.2011.032
DOI:10.3842/SIGMA.2011.032
Посилання:
Quantum Integrable Model of an Arrangement of Hyperplanes / A. Varchenko // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 29 назв. — англ.
Підтримка:
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
Дата:
2011
Переглядів:
496
Завантажень:
227
Анотація:
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.
