Quasigraded bases in loop algebras and classical rational r-matrices

In the present paper we construct quasigraded bases in the loop algebras associated with each classical rational r-matrix. We show that they are orthogonal and agreed with the decomposition of the loop algebra into sum of subalgebras that corresponds to this rational r-matrix. Using the quasigraded character of the constructed algebras we define the infinite sequence of the embedded ideals in the each subalgebra of the decomposition and apply this result to the theory of finite-dimensional integrable systems. © 2012 American Institute of Physics.