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Decompositions of quasigraded Lie algebras, non-skew-symmetric classical \(r\)-matrices and generalized Gaudin models. (English) Zbl 1284.17017

Summary: 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.

MSC:

17B80 Applications of Lie algebras and superalgebras to integrable systems
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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