Invertible linear transformations and the Lie algebras |
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Institution: | 1. Mathematical School, Liaoning Normal University, Dalian 116029, PR China;2. Department of Computer Sciences, Hong Kong Baptist University, Hong Kong, PR China;3. Information School, Shandong University of Science and Technology, Qingdao Huangdao 266510, PR China;1. School of Energy Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan Province 611731, PR China;2. School of Civil and Environmental Engineering, University of New South Wales, Sydney, NSW 2052, Australia |
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Abstract: | With the help of invertible linear transformations and the known Lie algebras, a way to generate new Lie algebras is given. These Lie algebras obtained have a common feature, i.e. integrable couplings of solitary hierarchies could be obtained by using them, specially, the Hamiltonian structures of them could be worked out. Some ways to construct the loop algebras of the Lie algebras are presented. It follows that some various loop algebras are given. In addition, a few new Lie algebras are explicitly constructed in terms of the classification of Lie algebras proposed by Ma Wen-Xiu, which are bases for obtaining new Lie algebras by using invertible linear transformations. Finally, some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy are obtained, whose Hamiltonian form-expressions are manifested by using the quadratic-form identity. |
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