r-Matrices for integrable systems

We consider some algebraic and geometric aspects of the theory of integrable systems in finite dimensions, associated with the existence of a classical $r$-matrix, first introduced by Sklyanin as the classical analogue of the quantum version. The importance of the notion of the $r$-matrix in this context relies on the fact that it connects the Hamiltonian structure of integrable equations with the factorisation problem which provides their explicit solution. In this framework, the Lax matrix is interpreted as the coadjoint orbit of a Lie algebra $\mathfrak{g}$, and the existence of a non-dynamical $r$-matrix allows the introduction of a second Lie algebra structure on $\mathfrak{g}$. Depending on the properties of the $r$-matrix associated with the specific system, we distinguish between bialgebras and dialgebras. Bialgebras are associated with a skew-symmetric $r$-matrix, were introduced by Drinfeld, and connected to the interplay between the two Lie algebras structures on $\mathfrak{g}$ and its dual $\mathfrak{g}^*$ respectively. Dialgebras refer to a larger class of $r$-matrix and are related to the factorisation properties of the system, were introduced by Semenov-Tian-Shansky and consist in two Lie algebras $\mathfrak{g}$ and $\mathfrak{g}_R$ coexisting on the same vector space.