Separation of variables for linear Lax algebras and classical r-matrices
Separation of variables for linear Lax algebras and classical r-matrices
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线性 Lax 代数和经典 r 矩阵的变量分离
DOI：
10.1063/1.5031769
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发表时间：
2018
影响因子：
1.3
通讯作者：
T. Skrypnyk
中科院分区：
文献类型：
--
作者：
B. Dubrovin;T. Skrypnyk
研究了代数可积Hamilton系统的分离变量问题，该系统具有依赖于谱参数的gl（n）-值Lax矩阵，且该矩阵满足线性Poisson括号，且gl（n）-值经典r-矩阵满足线性Poisson括号.我们制定，在相应的r-矩阵，一个充分条件，保证了“分离多项式”Sklyanin [Commun. Math.Phys.150，181（1992）]、Scott [J.Math.Phys.35，5831（1994）]、Gekhtman [Commun. Math.Phys.167，593（1995）]，以及Diener和Dubrovin（R-矩阵形式中的代数-几何达布坐标，西萨，预印本报告No.88 -94-FM，1994）产生了一个典型变量系统。我们考虑两个经典的r-矩阵和分离多项式的例子。这些例子之一，即，Skrypnyk [Phys. Lett. A 334，390（2005）; 347，266（2005）]和相应的分离多项式是新的。我们证明了Diener和Dubrovin的分离多项式在这种情况下对相应的有或无外磁场的广义Gaudin模型都产生一个完全的分离变量集.我们考虑了代数可积Hamilton系统的分离变量问题，该系统具有依赖于谱参数的gl（n）值Lax矩阵，且满足线性Poisson括号，其中gl（n）≥ gl（n）-赋值经典r-矩阵我们制定，在相应的r-矩阵，一个充分条件，保证了“分离多项式”Sklyanin [Commun. Math.Phys.150，181（1992）]、Scott [J.Math.Phys.35，5831（1994）]、Gekhtman [Commun. Math.Phys.167，593（1995）]，以及Diener和Dubrovin（R-矩阵形式中的代数-几何达布坐标，西萨，预印本报告No.88 -94-FM，1994）产生了一个典型变量系统。我们考虑两个经典的r-矩阵和分离多项式的例子。这些例子之一，即，Skrypnyk [Phys. Lett. A 334，390（2005）; 347，266（2005）]和相应的分离多项式是新的。我们展示了……
We consider the problem of separation of variables for the algebraically integrable Hamiltonian systems possessing gl(n)-valued Lax matrices depending on a spectral parameter that satisfy linear Poisson brackets with some gl(n) ⊗ gl(n)-valued classical r-matrices. We formulate, in terms of the corresponding r-matrices, a sufficient condition that guarantees that the “separating polynomials” of Sklyanin [Commun. Math. Phys. 150, 181 (1992)], Scott [J. Math. Phys. 35, 5831 (1994)], Gekhtman [Commun. Math. Phys. 167, 593 (1995)], and Diener and Dubrovin (Algebraic-geometrical Darboux coordinates in R-matrix formalism, SISSA, Preprint Report No. 88-94-FM, 1994) produce a system of canonical variables. We consider two examples of classical r-matrices and separating polynomials. One of these examples, namely, the n-parametric family of non-skew-symmetric non-dynamical classical r-matrices of Skrypnyk [Phys. Lett. A 334, 390 (2005); 347, 266 (2005)] and the corresponding separating polynomials is new. We show that the separating polynomials of Diener and Dubrovin produce in this case a complete set of separated variables for the corresponding generalized Gaudin models with or without external magnetic field.We consider the problem of separation of variables for the algebraically integrable Hamiltonian systems possessing gl(n)-valued Lax matrices depending on a spectral parameter that satisfy linear Poisson brackets with some gl(n) ⊗ gl(n)-valued classical r-matrices. We formulate, in terms of the corresponding r-matrices, a sufficient condition that guarantees that the “separating polynomials” of Sklyanin [Commun. Math. Phys. 150, 181 (1992)], Scott [J. Math. Phys. 35, 5831 (1994)], Gekhtman [Commun. Math. Phys. 167, 593 (1995)], and Diener and Dubrovin (Algebraic-geometrical Darboux coordinates in R-matrix formalism, SISSA, Preprint Report No. 88-94-FM, 1994) produce a system of canonical variables. We consider two examples of classical r-matrices and separating polynomials. One of these examples, namely, the n-parametric family of non-skew-symmetric non-dynamical classical r-matrices of Skrypnyk [Phys. Lett. A 334, 390 (2005); 347, 266 (2005)] and the corresponding separating polynomials is new. We show th...