Lie Algebras of Heat Operators in a Nonholonomic Frame

V. M. Buchstaber ; E. Yu. Bunkova

Matematičeskie zametki, Tome 108 (2020) no. 1, pp. 17-32.

Résumé

We construct the Lie algebras of systems of $2g$ graded heat operators $Q_0,Q_2,\dots,Q_{4g-2}$ that determine the sigma functions $\sigma(z,\lambda)$ of hyperelliptic curves of genera $g=1$, $2$, and $3$. As a corollary, we find that the system of three operators $Q_0$, $Q_2$, and $Q_4$ is already sufficient for determining the sigma functions. The operator $Q_0$ is the Euler operator, and each of the operators $Q_{2k}$, $k>0$, determines a $g$-dimensional Schrödinger equation with potential quadratic in $z$ for a nonholonomic frame of vector fields in the space $\mathbb C^{2g}$ with coordinates $\lambda$. For any solution $\varphi(z,\lambda)$ of the system of heat equations, we introduce the graded ring $\mathscr R_\varphi$ generated by the logarithmic derivatives of $\varphi(z,\lambda)$ of order $\ge 2$ and present the Lie algebra of derivations of $\mathscr R_\varphi$ explicitly. We show how this Lie algebra is related to our system of nonlinear equations. For $\varphi(z,\lambda)=\sigma(z,\lambda)$, this leads to a well-known result on how to construct the Lie algebra of differentiations of hyperelliptic functions of genus $g=1,2,3$.

Détail

Keywords: heat operator, grading, differentiation of Abelian functions over parameters.
Mots-clés : polynomial Lie algebra

Bibliographie

[1] V. M. Bukhshtaber, D. V. Leikin, “Uravneniya teploprovodnosti v negolonomnom repere”, Funkts. analiz i ego pril., 38:2 (2004), 12–27 | DOI | MR | Zbl

[2] V. M. Buchstaber, V. Z. Enolskii, D. V. Leikin, Kleinian Functions, Hyperelliptic Jacobians and Applications, Reviews in Math. and Math. Phys., 10, Gordon and Breach, London, 1997

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

[4] V. M. Bukhshtaber, “Polinomialnye dinamicheskie sistemy i uravnenie Kortevega–de Friza”, Sovremennye problemy matematiki, mekhaniki i matematicheskoi fiziki. II, Tr. MIAN, 294, MAIK «Nauka/Interperiodika», M., 2016, 191–215 | DOI | MR

[5] V. I. Arnold, Osobennosti kaustik i volnovykh frontov, Fazis, M., 1996 | MR

[6] V. M. Bukhshtaber, A. V. Mikhailov, “Beskonechnomernye algebry Li, opredelyaemye prostranstvom simmetricheskikh kvadratov giperellipticheskikh krivykh”, Funkts. analiz i ego pril., 51:1 (2017), 4–27 | DOI | MR | Zbl

[7] E. Yu. Bunkova, “Differentiation of genus 3 hyperelliptic functions”, Eur. J. Math., 4:1 (2018), 93–112 ; arXiv: 1703.03947 | DOI | MR | Zbl

[8] J. C. Eilbeck, J. Gibbons, Y. Onishi, S. Yasuda, Theory of Heat Equations for Sigma Functions, 2018, arXiv: 1711.08395

[9] V. M. Bukhshtaber, D. V. Leikin, “Differentsirovanie abelevykh funktsii po parametram”, UMN, 62:4 (376) (2007), 153–154 | DOI | MR | Zbl

[10] V. M. Bukhshtaber, D. V. Leikin, “Reshenie zadachi differentsirovaniya abelevykh funktsii po parametram dlya semeistv $(n,s)$-krivykh”, Funkts. analiz i ego pril., 42:4 (2008), 24–36 | DOI | MR | Zbl

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

[12] D. V. Millionschikov, “Polinomialnye algebry Li i rost ikh konechno porozhdennykh podalgebr Li”, Topologiya i fizika, Tr. MIAN, 302, MAIK «Nauka/Interperiodika», M., 2018, 316–333 | DOI | MR

[13] J. D. Cole, “On a quasi-linear parabolic equation occurring in aerodynamics”, Quart. Appl. Math., 9:3 (1951), 225–236 | DOI | MR | Zbl

[14] E. Hopf, “The partial differential equation $u_t+uu_x=u_{xx}$”, Comm. Pure Appl. Math., 3 (1950), 201–230 | DOI | MR | Zbl

[15] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov: metod obratnoi zadachi, Nauka, M., 1980 | MR

[16] L. Stickelberger, F. G. Frobenius, “Über die Differentiation der elliptischen Functionen nach den Perioden und Invarianten”, J. Reine Angew. Math., 92 (1882), 311–327 | MR

[17] V. M. Buchstaber, V. Z. Enolski, D. V. Leykin, “Multi-variable sigma-functions: old and new results”, Integrable Systems and Algebraic Geometry, Vol. 2, LMS Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2019 (to appear); arXiv: 1810.11079

[18] B. A. Dubrovin, S. P. Novikov, “Periodicheskaya zadacha dlya uravnenii Kortevega–de Friza i Shturma–Liuvillya. Ikh svyaz s algebraicheskoi geometriei”, Dokl. AN SSSR, 219:3 (1974), 531–534 | MR | Zbl

[19] V. M. Buchstaber, V. Z. Enolskii, D. V. Leikin, Multi-Dimensional Sigma-Functions, 2012, arXiv: 1208.0990

[20] H. F. Baker, “On the hyperelliptic sigma functions”, Amer. J. Math., 20:4 (1898), 301–384 | DOI | MR | Zbl

[21] E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge, 1996 | MR | Zbl

[22] E. Yu. Bunkova, “On the problem of differentiation of hyperelliptic functions”, Eur. J. Math., 5:3 (2019), 712–719 ; arXiv: 1812.04245 | DOI | MR | Zbl