## Characteristic determinant and Manakov triple for the double elliptic integrable system

A. Grekov, A. Zotov

SciPost Phys. 10, 055 (2021) · published 4 March 2021

- doi: 10.21468/SciPostPhys.10.3.055
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### Abstract

Using the intertwining matrix of the IRF-Vertex correspondence we propose a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system. More precisely, it is a ratio of the normally ordered determinants, which turns into a single determinant in the classical case. With its help we reproduce the recently suggested expression for the eigenvalues of the Hamiltonians for the dual to elliptic Ruijsenaars model. Next, we study the classical counterpart of our construction, which gives expression for the spectral curve and the corresponding $L$-matrix. This matrix is obtained explicitly as a weighted average of the Ruijsenaars and/or Sklyanin type Lax matrices with the weights as in the theta function series definition. By construction the $L$-matrix satisfies the Manakov triple representation instead of the Lax equation. Finally, we discuss the factorized structure of the $L$-matrix.

### Cited by 3

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1}A. Grekov, -
^{2}Andrei Zotov

^{1}Stony Brook University [SUNY Stony Brook]^{2}Математический институт им. В. А. Стеклова / Steklov Mathematical Institute

- Russian Academy of Sciences (through Organization: Российская академия наук / Russian Academy of Science [RAS])
- Russian Science Foundation