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

A new Lie algebra G of the Lie algebra sl(2) is constructed with complex entries whose structure constants are real and imaginary numbers.A loop algebra G corresponding to the Lie algebra G is constructed,for which it is devoted to generating a soliton hierarchy of evolution equations under the framework of generalized zero curvature equation which is derived from the compatibility of the isospectral problems expressed by Hirota operators.Finally,we decompose 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,a Liouville integrable soliton hierarchy is obtained,furthermore,we obtain its bi-Hamiltonian structure by employing the quadratic-form identity.