[Paper Review] Commutative Poisson subalgebras for the Sklyanin bracket and deformations of known integrable models

В. В. Соколов, A. V. Tsiganov|ArXiv.org|Dec 8, 2001

Nonlinear Waves and Solitons7 references3 citations

TL;DR

This paper constructs new families of commutative Poisson subalgebras for the Sklyanin bracket, generating integrals in involution that deform known integrable systems such as the Goryachev-Chaplygin top, Toda lattice, and Heisenberg spin chain. Using polynomial and rational deformations of the Hamiltonian, the authors derive explicit integrable models with higher-degree or rational terms, preserving separation of variables while modifying the separated curves.

ABSTRACT

A hierarchy of commutative Poisson subalgebras for the Sklyanin bracket is proposed. Each of the subalgebras provides a complete set of integrals in involution with respect to the Sklyanin bracket. Using different representations of the bracket, we find some integrable models and a separation of variables for them. The models obtained are deformations of known integrable systems like the Goryachev-Chaplygin top, the Toda lattice and the Heisenberg model.

Motivation & Objective

• To construct new N-dimensional commutative Poisson subalgebras $\mathfrak{A}_N^M$ for the Sklyanin bracket, generalizing the standard trace subalgebra $\mathfrak{A}_N^0$.
• To provide explicit deformations of known integrable systems—such as the Goryachev-Chaplygin top, Toda lattice, and Heisenberg spin chain—using higher-degree polynomial and rational functions in the Hamiltonian.
• To preserve the same canonical separation of variables as the original systems while modifying the separated curves, ensuring integrability.
• To demonstrate that these deformations yield new integrable models with additional cubic integrals of motion, even on special Casimir level sets.

Proposed method

• The construction relies on a $2\times2$ matrix $T(\lambda)$ with entries that are polynomials in $\lambda$, whose coefficients generate a Poisson algebra under the Sklyanin bracket.
• New commutative subalgebras $\mathfrak{A}_N^M$ are generated by $N$ integrals in involution $I_1^M, \dots, I_N^M$, with generators being polynomials of degree $M$ in the coefficients of $T(\lambda)$.
• Separation of variables is performed using canonical variables derived from the Goryachev-Chaplygin top, with the same $q_{1,2}, p_{1,2}$ as in the original system.
• The method applies to different representations of the Sklyanin bracket: on $e(3)$ for the rigid body, on $\mathbb{R}^{2N}$ for the Toda lattice, and on $sl^*(2)^N$ for the spin chain.
• Deformations are introduced via additional parameters $a_m, b_m, c_m$ in the generating polynomials, leading to Hamiltonians that are quadratic or rational in momenta and coordinates.
• The integrability is verified by showing that the deformed Hamiltonians commute with additional cubic integrals, especially on the Casimir level $(x,J)=0$.

Experimental results

Research questions

• RQ1 Can new commutative Poisson subalgebras of higher-degree polynomials be constructed for the Sklyanin bracket, beyond the standard trace subalgebra?
• RQ2 Do these new subalgebras yield integrable deformations of known systems like the Goryachev-Chaplygin top and Toda lattice?
• RQ3 Can the same separation of variables be preserved while modifying the separated curves in the deformed models?
• RQ4 What is the explicit form of the deformed Hamiltonians, and how do they differ from the original integrable systems?
• RQ5 Are the deformed systems still integrable, and do they admit additional conserved quantities beyond the standard ones?

Key findings

• The paper constructs $N$-dimensional commutative Poisson subalgebras $\mathfrak{A}_N^M$ for the Sklyanin bracket, with generators being polynomials of degree $M$ in the matrix coefficients, generalizing the standard trace subalgebra $\mathfrak{A}_N^0$.
• For $M=1$, the deformed Hamiltonian of the Goryachev-Chaplygin top is explicitly given as $H = J_1^2 + J_2^2 + 4J_3^2 + 2c_1x_1 + 2c_2x_2 + c_3J_3 + 4(a_1x_1 + a_2x_2)J_3 - (a_1^2 + a_2^2)x_3^2$, which preserves a cubic integral of motion.
• The same canonical separation variables $q_{1,2}, p_{1,2}$ used in the original Goryachev-Chaplygin top are valid for the deformed system, but the separated curve is modified by the deformation parameters.
• For the Toda lattice, a quadratic deformation is derived as $I_{N-1} = -a_0\sum p_i - a_1(\sum_{i>j} p_i p_j + \sum e^{q_{i+1}-q_i}) + e^{q_1}(c_0 + c_1 p_1) - e^{-q_N}(b_0 + b_1 p_N)$, preserving integrability.
• For the spin chain, a quadratic deformation Hamiltonian is constructed as $H = a_0 S_3 + b_0 S_+ + c_0 S_- + a_1(\sum_{j>i}(s_+^i s_-^j + s_3^i s_3^j) - S_3^2) - 2b_1(S_3 S_+ - \sum_{j>i} s_3^i s_+^j + \frac{1}{2}\sum s_3^i s_+^i) - 2c_1(S_3 S_- - \sum_{j<i} s_3^i s_-^j + \frac{1}{2}\sum s_3^i s_-^i)$.
• A rational deformation of the Goryachev-Chaplygin top is given by $\tilde{H} = (H_1 - a_0(2J_3 - 1))/(2J_3)$, which corresponds to a non-polynomial but integrable modification of the original system.