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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
