Abstract
We study different algebraic and geometric properties of Heisenberg (H-) invariant Poisson polynomial algebras with 5 generators. These algebras are unimodular, and the elliptic Feigin-Odesskii-Sklyanin Poisson algebras \(q_{n,k}(Y)\) constitute the main important example. We discuss all the quadratic H-invariant Poisson tensors on \({\mathbb C}^5\). For the Sklyanin algebras \(q_{5,1}(Y)\) and \(q_{5;2}(Y)\), we explicitly write the Poisson morphisms on the moduli space of the vector bundles on the normal elliptic curve Y in \(\mathbb P^4\), studied by Polishchuk and Odesskii-Feigin as the quadro-cubic Cremona transformation on \(\mathbb P^4\).
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Notes
- 1.
I am grateful to Brent Pym for enlightened discussion of this point.
- 2.
During a discussion with Sasha Odesskii, we found out that he independently obtained the same result, but it is still unpublished.
- 3.
This means that the embedding of Y in \(\mathbb P^4\) is given by complete linear system.
- 4.
A ruled degree 5 surface in \(\mathbb P^4.\)
- 5.
In fact the entries of the Moore-like syzygy matrix depends on \(\imath (z)\) where \(\imath (z_i)=z_i\) is the involution. But we need the matrix \(L_{\lambda }(z)\) to describe the surfaces of symplectic foliations which are \(\imath -\)invariants (compare with [4]).
References
Artin, M., Schelter, W.F.: Graded algebras of global dimension \(3\). Adv. Math. 66(2), 171–216 (1987)
Atiyah, M.: Vector bundles over an elliptic curves. Proc. Lond. Math. Soc. VII(3), 414-452 (1957)
Auret, A., Decker, W., Hulek, K., Popesku, S., Ranestad, K.: The geometry of Bielliptic surfaces in \(\mathbb{P}^4\). Int. J. of Math. 4, 873–902 (1993)
Auret, A., Decker, W., Hulek, K., Popesku, S., Ranestad, K.: Syzygies of abelian and bielliptic surfaces in \(\mathbb{P}^4\). Int. J. Math. 08, 849 (1997)
Bart, W., Hulek, K., Moore, R.: Shioda modular surface \(S(5)\) and the Horrocks-Mumford bundle, Vector bundles on algebraic varieties. In: Papers presented at the Bombay colloquim 1984, pp. 35-106. Oxford University Press, Bombay (1987)
Bianchi, L.: Ueber die Normsalformen dritter und fünfter Stufe des Elliptischen Integrals Ersten Gattung. Math. Ann. 17, 234–262 (1880)
Crauder, B., Katz, S.: Cremona transformations with smooth irreducible fundamental locus. Am. J. Math. 111, 289–309 (1989)
Feĭgin, B.L., Odesskiĭ, A.V.: Sklyanin’s elliptic algebras and moduli of vector bundles on elliptic curves. RIMS Kyoto University preprint (1998)
Feĭgin, B.L., Odesskiĭ, A.V.: Vector bundles on an elliptic curve and Sklyanin algebras (Russian) 268(2), 285–287 (1988) (prepr. BITP, Kiev)
Feĭgin, B.L., Odesskiĭ, A.V.: Vector bundles on an elliptic curve and Sklyanin algebras. Topics in quantum groups and finite-type invariants. Am. Math. Soc. Transl. Ser. Am. Math. Soc. 185(2), 65–84 (1998) (Providence, RI)
Fisher, T.: Genus one curves defined by pfaffians 185(2), 65–84 (2006)
Fisher, T.: Invariant theory for the elliptic normal quintic I. Twist of \(X(5)\). Math. Ann. 356(2), 589–616 (2013)
Fisher, T.: The invariants of a genus 1 curve. Proc. Lond. Math. Soc. 97(3), 753–782 (2008)
Fisher, T.: Pfaffian presentation of elliptic normal curves. Trans. Am. Math. Soc. 362(5), 2525–2540 (2010)
Hulek, K.: Projective geometry of elliptic curves. SMF, Astérisque, vol.137 (1986). https://books.google.ru/books?id=BgioAAAAIAAJ
Hulek, K., Katz, S., Schreyer, F.-O.: Cremona transformations and syzygies. Math. Z. 209, 419–443 (1992)
Klein, F.: Vorlesungen über das Ikosaeder und die Auflösungen der Gleichungen von fünftem Grade. Kommentiert und herausgegeben von P. Slodowy, Birkhäuser (1992)
Moore, R.: Heisenberg-invariant quintic 3-folds and sections of the Horrocks-Mumford bundle. Research Report No. 33-1985, Department of Mathematics, University of Canberra
Nambu, Yo.: Generalized Hamiltonian dynamics. Phys. Rev. D 7(3), 2405–2412 (1973)
Odesskiĭ, A.V., Feĭgin, B.L.: Sklyanin’s elliptic algebras. (Russian) Funktsional. Anal. i Prilozhen. 23(3), 45–54 (1989). (Translation in Funct. Anal. Appl.23(3), 207–214, 1989)
Odesskiĭ, A.V., Rubtsov, V.N.: Integrable systems associated with elliptic algebras. Quantum groups, 81–105. (IRMA Lect. Math. Theor. Phys., 12, Eur. Math. Soc. Zurich 2008)
Odesskiĭ, A.V., Rubtsov, V.N.: Polynomial Poisson algebras with a regular structure of symplectic leaves. (Russian) Teoret. Mat. Fiz. 133(1), 3–23 (2002)
Odesskiĭ, A.V.: Rational degeneration of elliptic quadratic algebras. Infinite analysis, Part A, B, Kyoto, pp. 773–779 (1991). (Adv. Ser. Math. Phys. 16, World Sci. Publ., River Edge, NJ, 1992)
Odesskii, A.V.: Elliptic algebras. Russ. Math. Surv. 57(6), 1127–1162 (2002)
Odesskii, A.V.: Bihamiltonian elliptic structures. Mosc. Math. J. 982(4), 941–946 (2004)
Ortenzi G., Rubtsov, V., Tagne Pelap, S.R.: On the Heisenberg invariance and the elliptic Poisson tensors. Lett. Math. Phys. 96(1–3), 263–284 (2011)
Ortenzi, G., Rubtsov, V., Tagne Pelap, S.R.: Integer solutions of integral inequalities and \(H\)-invariant Jacobian Poisson structures. Adv. Math. Phys. 2011, 252186 (2011)
Polishchuk, A.: Algebraic geometry of Poisson brackets. J. Math. Sci. 84, 1413–1444 (1997)
Polishchuk, A.: Poisson structures and birational morphisms associated with bundles on elliptic curves. Int. Math. Res. Notes 13, 683–703 (1998)
Pym, B.: Constructions and classifications of projective Poisson varieties. arXiv:1701.08852
Semple, J.G., Roth, L.: Projective algebraic geometry. Oxford University Press (1986)
Semple, J.G.: Cremona transformations of space of four dimensions by means of quadrics and the reverse transformations. Phil. Trans. R. Soc. Lond. Ser. A. 228, 331–376 (1929)
Sklyanin, E.K.: Some algebraic structures connected with the Yang-Baxter equation. (Russian) Funktsional. Anal. i Prilozhen. 16(4), 27–34 (1982)
Sklyanin, E.K.: Some algebraic structures connected with the Yang-Baxter equation. Representations of a quantum algebra. (Russian) Funktsional. Anal. i Prilozhen. 17(4), 34–48 (1983)
Smith, S.P., Stafford, J.T.: Regularity of the four-dimensional Sklyanin algebra. Compositio Math. 83(3), 259–289 (1992)
Tagne Pelap, S.R.: Poisson (co)homology of polynomial Poisson algebras in dimension four: Sklyanin’s case. J. Algebra 322(4), 1151–1169 (2009)
Tagne Pelap, S.R.: On the Hochschild homology of elliptic Sklyanin algebras. Lett. Math. Phys. 87(4), 267–281 (2009)
Takhtajan, Leon: On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160(2), 295–315 (1994)
Tate, J., Van den Bergh, M.: Homological properties of Sklyanin algebras. Invent. Math. 124(1–3), 619–647 (1996)
Tu, L.W.: Semistable bundles over an elliptic curve. Adv. Math. 98(1), 1–26 (1993)
Hua, Z., Polishchuk, A.: Shifted Poisson structures and moduli spaces of complexes. arXiv: math:1706.09965
Acknowledgements
During this work the author benefited from many useful discussions and suggestions of many colleagues. He is thankful to Brent Pym with whom he discussed the holomorphic Poisson geometry, to Sasha Polishchuk for his help with Algebraic Geometry and to Alexander Odesskii who taught him the Elliptic Algebras. Igor Reider had clarified some question and helped to understand the relation with his own unpublished results.
Some parts of this paper are based on previously published results of the author in a collaboration with Giovanni Ortenzi and Serge Tagne Pelap as well as with A. Odesskii. He is grateful to them for their collaboration.
A special thank to Rubik Pogossyan who had verified with Mathematica Package the author’s hint statement about the form and exact value of the determinant of the Jacobian for cubic “Inverse” Cremona map.
He had benefited from a lot of numerous conversations with Boris Feigin and Alexei Gorodentsev on various related subjects.
Finally, this work would have never been written without two author’s talks on the Moscow HSE Bogomolov Laboratory Seminar in June 2016 and on the 1st International Conference of Mathematical Physics at Kezenoy-Am in November 2016. He greatly acknowledges the invitation of Misha Verbitsky to Moscow and of V. Buchstaber and A. Mikhailov to the Chechen Republic. His special thanks to Dima Grinev and Sotiris Konstantinou-Rizos for their hospitality, excellent organisation of the Conference at Kezenoy-Am and inspiration. He also thanks Sotiris Konstantinou-Rizos for his great patience and valuable help while the text was being prepared to submission.
His work was partly supported by the Russian Foundation for Basic Research (Projects 18-01-00461 and 16-51-53034-GFEN). Part of this work was carried out within the framework of the State Programme of the Ministry of Education and Science of the Russian Federation, project 1.12873.2018/12.1.
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Rubtsov, V. (2018). Quadro-Cubic Cremona Transformations and Feigin-Odesskii-Sklyanin Algebras with 5 Generators. In: Buchstaber, V., Konstantinou-Rizos, S., Mikhailov, A. (eds) Recent Developments in Integrable Systems and Related Topics of Mathematical Physics. MP 2016. Springer Proceedings in Mathematics & Statistics, vol 273. Springer, Cham. https://doi.org/10.1007/978-3-030-04807-5_6
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