Reduction in soliton hierarchies and special points of classical<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi>r</mml:mi></mml:math>-matrices

Skrypnyk, T.

doi:10.1016/j.geomphys.2018.03.023

Journal of Geometry and Physics

2018

DOI: 10.1016/j.geomphys.2018.03.023

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Reduction in soliton hierarchies and special points of classicalr-matrices

T. Skrypnyk

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2022

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On the general solution of the permuted classical Yang–Baxter equation and quasigraded Lie algebras

Skrypnyk

2022

Journal of Mathematical Physics

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Using the technique of the quasigraded Lie algebras, we construct general spectral-parameter dependent solutions r12( u, v) of the permuted classical Yang–Baxter equation with the values in the tensor square of simple Lie algebra [Formula: see text]. We show that they are connected with infinite-dimensional Lie algebras with Adler–Kostant–Symmes decompositions and are labeled by solutions of a constant quadratic equation on the linear space [Formula: see text], N ≥ 1. We formulate the conditions when the corresponding r-matrices are skew-symmetric, i.e., they are equivalent to the ones described by Belavin–Drinfeld classification. We illustrate the developed theory by the example of the elliptic r-matrix of Sklyanin. We apply the obtained result to the explicit construction of the generalized quantum and classical Gaudin spin chains.

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On the general solution of the permuted classical Yang–Baxter equation and quasigraded Lie algebras

Skrypnyk

2022

Journal of Mathematical Physics

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show abstract

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Reduction in soliton hierarchies and special points of classicalr-matrices

Cited by 1 publication

References 29 publications

On the general solution of the permuted classical Yang–Baxter equation and quasigraded Lie algebras

On the general solution of the permuted classical Yang–Baxter equation and quasigraded Lie algebras

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