Decompositions of quasigraded Lie algebras, non-skew-symmetric classical r-matrices and generalized Gaudin models

We construct a special family of quasigraded Lie algebras that generalize loop algebras in different gradings and admit Adler-Kostant-Symes decomposition into a sum of two subalgebras. We analyze the special cases when the constructed Lie algebras admit additionally other types of Adler-Kostant-Symes decompositions. Based on the proposed Lie algebras and their decompositions we explicitly construct several new classes of non-skew-symmetric classical r-matrices r(u,v) with spectral parameters. Using them we obtain new types of the generalized quantum and classical Gaudin spin chains. © 2013 Elsevier B.V