Decompositions of quasigraded Lie algebras, non-skew-symmetric classical rr-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 rr-matrices r(u,v)r(u,v) with spectral parameters. Using them we obtain new types of the generalized quantum and classical Gaudin spin chains.