Taras Skrypnyk (University of Milano-Bicocca, Italy)

Classical r-matrices, infinite-dimensional Lie algebras and integrable hierarchies of Landau–Lifsitz type

Abstract:
For each finite-dimensional simple Lie algebra $\mathfrak{g}$, starting from a general $\mathfrak{g}\otimes \mathfrak{g}$-valued solutions $r(u,v)$ of the generalized classical Yang–Baxter equation, we construct ''$N$-poled'' infinite-dimensional Lie algebras $\widetilde{\mathfrak{g}}^{-,N}_r$ of $\mathfrak{g}$-valued meromorphic functions with the poles in a fixed sets of points $\nu_1,\dots,\nu_N$. We apply the constructed algebras to the theory of soliton equations and obtain with their help the most general form of integrable Landau–Lifshitz-type and anisotropic chiral field-type equations.