Geodesic
Quadratic algebras related to elliptic curves
Teoretičeskaâ i matematičeskaâ fizika, Tome 156 (2008) no. 2, pp. 163-183

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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.
Mots-clés : Poisson structure
Keywords: integrable system. 
@article{TMF_2008_156_2_a0,
     author = {A. V. Zotov and A. M. Levin and M. A. Olshanetsky and Yu. B. Chernyakov},
     title = {Quadratic algebras related to elliptic curves},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {163--183},
     publisher = {mathdoc},
     volume = {156},
     number = {2},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2008_156_2_a0/}
}
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A. V. Zotov; A. M. Levin; M. A. Olshanetsky; Yu. B. Chernyakov. Quadratic algebras related to elliptic curves. Teoretičeskaâ i matematičeskaâ fizika, Tome 156 (2008) no. 2, pp. 163-183. http://geodesic.mathdoc.fr/item/TMF_2008_156_2_a0/