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

For each finite-dimensional simple Lie algebra g, starting from a general g ⊗ g-valued solutions r(u, v) of the generalized classical Yang-Baxter equation, we construct infinite-dimensional Lie algebras g̃r- of g-valued meromorphic functions. We outline two ways of embedding of the Lie algebra g̃r- into a larger Lie algebra with Kostant-Adler-Symmes decomposition. The first of them is an embedding of g̃r- into Lie algebra g̃(u-1, u) of formal Laurent power series. The second is an embedding of g̃r- as a quasigraded Lie subalgebra into a quasigraded Lie algebra g̃r: g̃r + g̃r, such that the Kostant-Adler-Symmes decomposition is consistent with a chosen quasigrading. We construct dual spaces g̃r*, (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 g̃r, 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 g. © 2013 AIP Publishing LLC.