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Mat. Zametki, 2020, Volume 108, Issue 1, Pages 17–32 (Mi mz12791)  

Lie Algebras of Heat Operators in a Nonholonomic Frame

V. M. Buchstaber, E. Yu. Bunkova

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: We construct the Lie algebras of systems of $2g$ graded heat operators $Q_0,Q_2,…,Q_{4g-2}$ 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 $g=1,2,3$.

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

DOI: https://doi.org/10.4213/mzm12791

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English version:
Mathematical Notes, 2020, 108:1, 15–28

Bibliographic databases:

UDC: 517.986
Received: 28.10.2019
Revised: 13.02.2020

Citation: V. M. Buchstaber, E. Yu. Bunkova, “Lie Algebras of Heat Operators in a Nonholonomic Frame”, Mat. Zametki, 108:1 (2020), 17–32; Math. Notes, 108:1 (2020), 15–28

Citation in format AMSBIB
\Bibitem{BucBun20}
\by V.~M.~Buchstaber, E.~Yu.~Bunkova
\paper Lie Algebras of Heat Operators in a Nonholonomic Frame
\jour Mat. Zametki
\yr 2020
\vol 108
\issue 1
\pages 17--32
\mathnet{http://mi.mathnet.ru/mz12791}
\crossref{https://doi.org/10.4213/mzm12791}
\transl
\jour Math. Notes
\yr 2020
\vol 108
\issue 1
\pages 15--28
\crossref{https://doi.org/10.1134/S0001434620070020}
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