Classical R-Operators and Integrable Generalizations of Thirring Equations

We construct different integrable generalizations of the massive Thirring equations corresponding loop algebras $\widetilde{\mathfrak{g}}^{\sigma}$ in different gradings and associated ''triangular'' $R$-operators. We consider the most interesting cases connected with the Coxeter automorphisms, second order automorphisms and with ''Kostant-Adler-Symes'' $R$-operators. We recover a known matrix generalization of the complex Thirring equations as a partial case of our construction.Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Twisted rational r-matrices and algebraic Bethe ansatz: Application to generalized Gaudin and Richardson models

In the present paper we develop the algebraic Bethe ansatz approach to the case of non-skew-symmetric gl(2) circle times gl(2)-valued Cartan-non-invariant classical r-matrices with spectral parameters. We consider the two families of these r-matrices, namely, the two non-standard rational r-matrices twisted with the help of second order automorphisms and realize the algebraic Bethe ansatz method for them. We study physically important examples of the Gaudin-type and BCS-type systems associated with these r-matrices and obtain explicitly the Bethe vectors and the spectrum for the corresponding quantum hamiltonians in terms of solutions of Bethe equations. (C) 2021 The Author(s). Published by Elsevier B.V.0117U000240info:eu-repo/semantics/publishedVersio

"Doubled" generalized Landau-Lifshiz hierarchies and special quasigraded Lie algebras

Using special quasigraded Lie algebras we obtain new hierarchies of integrable nonlinear vector equations admitting zero-curvature representations. Among them the most interesting is extension of the generalized Landau-Lifshitz hierarchy which we call "doubled" generalized Landau-Lifshiz hierarchy. This hierarchy can be also interpreted as an anisotropic vector generalization of "modified" Sine-Gordon hierarchy or as a very special vector generalization of so(3) anisotropic chiral field hierarchy.Comment: 16 pages, no figures, submitted to Journal of Physics

Spin chains in magnetic field, non-skew-symmetric classical r-matrices and BCS-type integrable systems

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“Generalized” algebraic Bethe ansatz, Gaudin-type models and Z p -graded classical r -matrices

AbstractWe consider quantum integrable systems associated with reductive Lie algebra gl(n) and Cartan-invariant non-skew-symmetric classical r-matrices. We show that under certain restrictions on the form of classical r-matrices “nested” or “hierarchical” Bethe ansatz usually based on a chain of subalgebras gl(n)⊃gl(n−1)⊃...⊃gl(1) is generalized onto the other chains or “hierarchies” of subalgebras. We show that among the r-matrices satisfying such the restrictions there are “twisted” or Zp-graded non-skew-symmetric classical r-matrices. We consider in detail example of the generalized Gaudin models with and without external magnetic field associated with Zp-graded non-skew-symmetric classical r-matrices and find the spectrum of the corresponding Gaudin-type hamiltonians using nested Bethe ansatz scheme and a chain of subalgebras gl(n)⊃gl(n−n1)⊃gl(n−n1−n2)⊃gl(n−(n1+...+np−1)), where n1+n2+...+np=n