Separation of variables for type \(D_n\) Hitchin systems on a hyperelliptic curve. (English. Russian original) Zbl 1467.14088
Russ. Math. Surv. 76, No. 2, 363-365 (2021); translation from Usp. Mat. Nauk 76, No. 2, 181-182 (2021).
From the text: For Hitchin systems Darboux variables were until recently known only in the
case of genus 2 and rank 2 (see [K. Gawȩdzki and P. Tran-Ngoc-Bich, J. Math. Phys. 41, No. 7, 4695–4712 (2000; Zbl 0974.32009)]). For arbitrary simple Lie algebras a description
of the class of spectral curves for Hitchin systems on hyperelliptic curves of any
genus was given in [O. K. Sheinman, Funct. Anal. Appl. 53, No. 4, 291–303 (2019; Zbl 1445.14050); translation from Funkts. Anal. Prilozh. 53, No. 4, 63–78 (2019)]. For Lie algebras of types \(A_n\), \(B_n\), \(C_n\) Darboux variables
were found in an explicit form using separation of variables.
The goal of this paper is to find Darboux coordinates for Hitchin systems on a hyperelliptic curve for simple Lie algebras of type \(D_n\) on the basis of the explicit description of the spectral curve given in [loc. cit.]. Nevertheless, we were able to find Darboux coordinates explicitly only in the simplest case of the Lie algebra \(\mathfrak{so}(4)\).
The goal of this paper is to find Darboux coordinates for Hitchin systems on a hyperelliptic curve for simple Lie algebras of type \(D_n\) on the basis of the explicit description of the spectral curve given in [loc. cit.]. Nevertheless, we were able to find Darboux coordinates explicitly only in the simplest case of the Lie algebra \(\mathfrak{so}(4)\).
MSC:
14H70 | Relationships between algebraic curves and integrable systems |
37J37 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures |
References:
[1] | Esterov, A., Compos. Math., 155, 2, 229-245 (2019) · Zbl 1451.14152 · doi:10.1112/S0010437X18007868 |
[2] | Gawędzki, K.; Tran-Ngog-Bich, P., J. Math. Phys., 41, 7, 4695-4712 (2000) · Zbl 0974.32009 · doi:10.1112/S0010437X18007868 |
[3] | Hurtubise, J. C., Duke Math. J., 83, 1, 19-50 (1996) · Zbl 0857.58024 · doi:10.1215/S0012-7094-96-08302-7 |
[4] | Krichever, I., Comm. Math. Phys., 229, 2, 229-269 (2002) · Zbl 1073.14048 · doi:10.1007/s002200200659 |
[5] | Sheinman, O. K., Funktsional. Anal. Prilozhen., 53, 4, 63-78 (2019) · Zbl 1445.14050 · doi:10.4213/faa3648 |
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