Infinite-dimensional Lie algebras, classical r-matrices, and Lax operators: Two approaches

Abstract

For each finite-dimensional simple Lie algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {g}$\end{document}g, starting from a general \documentclass[12pt]{minimal}\begin{document}$\mathfrak {g}\otimes \mathfrak {g}$\end{document}g⊗g-valued solutions r(u, v) of the generalized classical Yang-Baxter equation, we construct infinite-dimensional Lie algebras \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{-}_r$\end{document}g̃r− of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {g}$\end{document}g-valued meromorphic functions. We outline two ways of embedding of the Lie algebra \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{-}_r$\end{document}g̃r− into a larger Lie algebra with Kostant-Adler-Symmes decomposition. The first of them is an embedding of \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{-}_r$\end{document}g̃r− into Lie algebra \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}(u^{-1},u))$\end{document}g̃(u−1,u)) of formal Laurent power series. The second is an embedding of \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{-}_r$\end{document}g̃r− as a quasigraded Lie subalgebra into a quasigraded Lie algebra \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}_r$\end{document}g̃r: \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}_r=\widetilde{\mathfrak {g}}^{-}_r+\widetilde{\mathfrak {g}}^{+}_r$\end{document}g̃r=g̃r−+g̃r+, such that the Kostant-Adler-Symmes decomposition is consistent with a chosen quasigrading. We construct dual spaces \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^*_r$\end{document}g̃r*, \documentclass[12pt]{minimal}\begin{document}$(\widetilde{\mathfrak {g}}^{\pm }_r)^*$\end{document}(g̃r±)* and explicit form of the Lax operators L(u), L±(u) as elements of these spaces. We develop a theory of integrable finite-dimensional hamiltonian systems and soliton hierarchies based on Lie algebras \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}_r$\end{document}g̃r, \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{\pm }_r$\end{document}g̃r±. We consider examples of such systems and soliton equations and obtain the most general form of integrable tops, Kirchhoff-type integrable systems, and integrable Landau-Lifshitz-type equations corresponding to the Lie algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {g}$\end{document}g.