A. V. Zotov, A. M. Levin, M. A. Olshanetsky, Yu. B. Chernyakov, “Quadratic algebras related to elliptic curves”, TMF, 156:2 (2008), 163–183; Theoret. and Math. Phys., 156:2 (2008), 1103

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Teoreticheskaya i Matematicheskaya Fizika, 2008, Volume 156, Number 2, Pages 163–183
DOI: https://doi.org/10.4213/tmf6238 (Mi tmf6238)

This article is cited in 16 scientific papers (total in 16 papers)

Quadratic algebras related to elliptic curves

A. V. Zotov^ab, A. M. Levin^bc, M. A. Olshanetsky^a, Yu. B. Chernyakov^a

^a Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)
^b Max Planck Institute for Mathematics
^c P. P. Shirshov institute of Oceanology of RAS

Full-text PDF (558 kB) Citations (16)

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DOI: https://doi.org/10.4213/tmf6238

Abstract: We construct quadratic finite-dimensional Poisson algebras corresponding to a rank-

$N$ degree-one vector bundle over an elliptic curve with

$n$ marked points and also construct the quantum version of the algebras. The algebras are parameterized by the moduli of curves. For

$N=2$ and

$n=1$ , they coincide with Sklyanin algebras. We prove that the Poisson structure is compatible with the Lie–Poisson structure defined on the direct sum of

$n$ copies of

$sl(N)$ . The origin of the algebras is related to the Poisson reduction of canonical brackets on an affine space over the bundle cotangent to automorphism groups of vector bundles.

Keywords: Poisson structure, integrable system.

Received: 14.08.2007

English version:
Theoretical and Mathematical Physics, 2008, Volume 156, Issue 2, Pages 1103–1122
DOI: https://doi.org/10.1007/s11232-008-0081-0

Bibliographic databases:

Language: Russian

Citation: A. V. Zotov, A. M. Levin, M. A. Olshanetsky, Yu. B. Chernyakov, “Quadratic algebras related to elliptic curves”, TMF, 156:2 (2008), 163–183; Theoret. and Math. Phys., 156:2 (2008), 1103–1122

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Linking options:

https://www.mathnet.ru/eng/tmf6238

https://doi.org/10.4213/tmf6238

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This publication is cited in the following 16 articles:

E. Trunina, A. Zotov, “Lax equations for relativistic $\mathrm{G}\mathrm{L}(NM,\mathbb{C})$ Gaudin models on elliptic curve”, J. Phys. A, 55:39 (2022), 395202–31
A. Zabrodin, A. Zotov, “Field analogue of the Ruijsenaars–Schneider model”, JHEP, 2022:7 (2022), 23–51
I. A. Sechin, A. V. Zotov, “Quadratic algebras based on $SL(NM)$ elliptic quantum $R$ -matrices”, Theoret. and Math. Phys., 208:2 (2021), 1156–1164
Krasnov T. Zotov A., “Trigonometric Integrable Tops From Solutions of Associative Yang-Baxter Equation”, Ann. Henri Poincare, 20:8 (2019), 2671–2697
Zotov A., “Relativistic Elliptic Matrix Tops and Finite Fourier Transformations”, Mod. Phys. Lett. A, 32:32 (2017), 1750169
A. M. Levin, M. A. Olshanetsky, A. V. Zotov, “Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras, Calogero–Moser systems, and KZB equations”, Theoret. and Math. Phys., 188:2 (2016), 1121–1154
Levin A. Olshanetsky M. Zotov A., “Classical Integrable Systems and Soliton Equations Related To Eleven-Vertex R-Matrix”, Nucl. Phys. B, 887 (2014), 400–422
Aminov G. Arthamonov S. Smirnov A. Zotov A., “Rational TOP and Its Classical R-Matrix”, J. Phys. A-Math. Theor., 47:30 (2014), 305207
Levin A. Olshanetsky M. Zotov A., “Relativistic Classical Integrable Tops and Quantum R-Matrices”, J. High Energy Phys., 2014, no. 7, 012
Levin A. Olshanetsky M. Smirnov A. Zotov A., “Characteristic Classes of Sl(N, C)-Bundles and Quantum Dynamical Elliptic R-Matrices”, J. Phys. A-Math. Theor., 46:3 (2013), 035201
A. V. Zotov, A. V. Smirnov, “Modifications of bundles, elliptic integrable systems, and related problems”, Theoret. and Math. Phys., 177:1 (2013), 1281–1338
Rains E., Ruijsenaars S., “Difference Operators of Sklyanin and Van Diejen Type”, Commun. Math. Phys., 320:3 (2013), 851–889
Mironov A., Morozov A., Runov B., Zenkevich Y., Zotov A., “Spectral Duality Between Heisenberg Chain and Gaudin Model”, Lett. Math. Phys., 103:3 (2013), 299–329
Aminov G., Mironov A., Morozov A., Zotov A., “Three-Particle Integrable Systems with Elliptic Dependence on Momenta and Theta Function Identities”, Phys. Lett. B, 726:4-5 (2013), 802–808
Andrey M. Levin, Mikhail A. Olshanetsky, Andrey V. Smirnov, Andrei V. Zotov, “Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles”, SIGMA, 8 (2012), 095, 37 pp.
Andrei V. Zotov, “ $1+1$ Gaudin Model”, SIGMA, 7 (2011), 067, 26 pp.

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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