Infinite series of Lie algebras and boundary conditions for integrable systems

We construct and study some new representations of the quadratic $R$-matrix algebras in classical and in quantum mechanics which are related to the Toda lattices associated with the classical simple Lie algebras. A new Lax representation for the Manakov top is presented. A dynamical $SO(2,1)$ algebra suited for the study of the adjoint Mathieu functions is constructed.