Geodesic
Infinite-Dimensional Lie Algebras Determined by the Space of Symmetric Squares of Hyperelliptic Curves
Funkcionalʹnyj analiz i ego priloženiâ, Tome 51 (2017) no. 1, pp. 4-27.

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We construct Lie algebras of vector fields on universal bundles of symmetric squares of hyperelliptic curves of genus $g=1,2,\dots$. For each of these Lie algebras, the Lie subalgebra of vertical fields has commuting generators, while the generators of the Lie subalgebra of projectable fields determines the canonical representation of the Lie subalgebra with generators $L_{2q}$, $q=-1, 0, 1, 2, \dots$, of the Witt algebra. As an application, we obtain integrable polynomial dynamical systems.
Keywords: infinite-dimensional Lie algebras, representations of the Witt algebra, symmetric polynomials, symmetric powers of curves, commuting operators, polynomial dynamical systems. 
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V. M. Buchstaber; A. V. Mikhailov. Infinite-Dimensional Lie Algebras Determined by the Space of Symmetric Squares of Hyperelliptic Curves. Funkcionalʹnyj analiz i ego priloženiâ, Tome 51 (2017) no. 1, pp. 4-27. http://geodesic.mathdoc.fr/item/FAA_2017_51_1_a1/

[1] V. I. Arnold, Osobennosti kaustik i volnovykh frontov, Biblioteka matematika, 1, FAZIS, M., 1996 | MR

[2] V. I. Arnold, “Wave front evolution and equivariant Morse lemma”, Comm. Pure Appl. Math., 29:6 (1976), 557–582 ; 30:6 (1977), 823, correction ; V. I. Arnold, Izbrannoe-60, FAZIS, M., 1997, 289–318 | DOI | MR | Zbl | DOI | MR | Zbl | MR

[3] V. M. Bukhshtaber, “Polinomialnye dinamicheskie sistemy i uravnenie Kortevega–de Friza”, Sovremennye problemy matematiki, mekhaniki i matematicheskoi fiziki, Sbornik statei, v. 2, Tr. MIAN, 294, MAIK, M., 2016, 191–215 | DOI

[4] V. M. Bukhshtaber, D. V. Leikin, “Polinomialnye algebry Li”, Funkts. anal. prilozhen., 36:4 (2002), 18–34 | DOI | MR | Zbl

[5] V. M. Buchstaber, V. Z. Enolskii, D. V. Leikin, “Hyperelliptic Kleinian functions and applications”, Solitons, Geometry and Topology: On the Crossroad, Amer. Math. Soc. Trans., Ser. 2, 179, Amer. Math. Soc., Providence, RI, 1997, 1–33 | DOI | MR

[6] B. Enriquez, V. Rubtsov, “Commuting families in skew fields and quantization of Beauville's fibration”, Duke Math. J., 119:2 (2003), 197–219 | DOI | MR | Zbl

[7] Ph. J. Higgins, K. Mackenzie, “Algebraic constructions in the category of the Lie algebroids”, J. Algebra, 129 (1990), 194–230 | DOI | MR | Zbl

[8] V. Kats, Beskonechnomernye algebry Li, Mir, M., 1995 | MR

[9] I. Makdonald, Simmetricheskie funktsii i mnogochleny Kholla, Mir, M., 1984 | MR

[10] K. Mackenzie, The General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, Cambridge, 2005 | MR | Zbl

[11] P. W. Michor, Topics in Differential Geometry, Graduate Studies in Math., 93, Amer. Math. Soc., Providence, RI, 2008 | DOI | MR | Zbl