Characteristic determinant and Manakov triple for the double elliptic integrable system

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.

Automated summary

This paper proposes a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system using the intertwining matrix of the IRF-Vertex correspondence. The representation reproduces the eigenvalues of the Hamiltonians for the dual to elliptic Ruijsenaars model and provides an expression for the spectral curve and L-matrix. The L-matrix is a weighted average of Lax matrices with weights from the theta function series definition, satisfying the Manakov triple representation instead of the Lax equation, and its factorized structure is discussed.