A Complex Higher-Dimensional Lie Algebra with Real and Imaginary Structure Constants as Well as Its Decomposition

摘要

A new Lie algebra G of the Lie algebra sl(2) is constructed with complex entries whose structure constantsare real and imaginary numbers.A loop algebra G corresponding to the Lie algebra G is constructed,for which itis devoted to generating a soliton hierarchy of evolution equations under the framework of generalized zero curvatureequation which is derived from the compatibility of the isospectral problems expressed by Hirota operators.Finally,wedecompose the Lie algebra G to obtain the subalgebras G_1 and G_2.Using the G_2 and its one type of loop algebra (?)_2,aLiouville integrable soliton hierarchy is obtained,furthermore,we obtain its bi-Hamiltonian structure by employing thequadratic-form identity.