SIGMA 5 (2009), 110, 22 pages      arXiv:0908.4064      https://doi.org/10.3842/SIGMA.2009.110
Contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions”

Manin Matrices, Quantum Elliptic Commutative Families and Characteristic Polynomial of Elliptic Gaudin Model

Vladimir Rubtsov ^{a, c}, Alexey Silantyev ^b and Dmitri Talalaev ^c
a) LAREMA, Université d'Angers, 2 Boulevard Lavoisier, 49045 Angers, France
^b) Department of Mathematics, University Gardens, University of Glasgow, G12 8QW, UK
^c) ITEP, B. Cheremushkinskaja 25, 117218 Moscow, Russia

Received March 30, 2009, in final form December 12, 2009; Published online December 24, 2009

Abstract
In this paper we construct the quantum spectral curve for the quantum dynamical elliptic gl_n Gaudin model. We realize it considering a commutative family corresponding to the Felder's elliptic quantum group E_τ,h(gl_n) and taking the appropriate limit. The approach of Manin matrices here suits well to the problem of constructing the generation function of commuting elements which plays an important role in SoV and Langlands concept.

Key words: Manin matrices; L-operators; elliptic Felder R-matrix; Gaudin models.

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