We first prove that for a finite dimensional non-semisimple Hopfalgebra H, the trivial H-module is not projective and so the almost split sequence ended with k exists. By this exact sequence, for all indecomposable H-...We first prove that for a finite dimensional non-semisimple Hopfalgebra H, the trivial H-module is not projective and so the almost split sequence ended with k exists. By this exact sequence, for all indecomposable H-module X, we can construct a special kind of exact sequence ending with it. The main aim of this paper is to determine when this special exact sequence is an almost split one. For this aim, we restrict H to be tmimodular and the square of its antipode to be an inner automorphism. As a special case, we give an application to the quantum double D（H）=（H^op）^＊∞ H） of any non-semisimple Hopf algebra.展开更多
We propose a class of non-semisimple matrix loop algebras consisting of 3×3 block matrices,and form zero curvature equations from the presented loop algebras to generate bi-integrable couplings.Applications are m...We propose a class of non-semisimple matrix loop algebras consisting of 3×3 block matrices,and form zero curvature equations from the presented loop algebras to generate bi-integrable couplings.Applications are made for the AKNS soliton hierarchy and Hamiltonian structures of the resulting integrable couplings are constructed by using the associated variational identities.展开更多
By introducing an invertible linear transform, a new Lie algebra G is obtained from the Lie algebra H. Making use of the compatibility conditions of the respective isospectral problems, a generalized NLS-MKdV hierarch...By introducing an invertible linear transform, a new Lie algebra G is obtained from the Lie algebra H. Making use of the compatibility conditions of the respective isospectral problems, a generalized NLS-MKdV hierarchy and a new integrable soliton hierarchy are achieved by using the trace identity or the variational identity. Then, two special non-semisimple Lie algebras ?and ?are explicitly conducted. As an application, the nonlinear continuous integrable couplings of the obtained integrable systems as well as their bi-Hamiltonian structures are established, respectively.展开更多
We propose a systematic approach for generating Hamiltonian tri-integrable couplings of soliton hierarchies. The resulting approach is based on semi-direct sums of matrix Lie algebras consisting of 4× 4 block mat...We propose a systematic approach for generating Hamiltonian tri-integrable couplings of soliton hierarchies. The resulting approach is based on semi-direct sums of matrix Lie algebras consisting of 4× 4 block matrix Lie algebras. We apply the approach to the AKNS soIRon hierarchy, and Hamiltonian structures of the obtained tri-integrable couplings are constructed by the variational identity.展开更多
基金Project (No. 10371107) supported by the National Natural ScienceFoundation of China
文摘We first prove that for a finite dimensional non-semisimple Hopfalgebra H, the trivial H-module is not projective and so the almost split sequence ended with k exists. By this exact sequence, for all indecomposable H-module X, we can construct a special kind of exact sequence ending with it. The main aim of this paper is to determine when this special exact sequence is an almost split one. For this aim, we restrict H to be tmimodular and the square of its antipode to be an inner automorphism. As a special case, we give an application to the quantum double D（H）=（H^op）^＊∞ H） of any non-semisimple Hopf algebra.
基金This work was supported by the Department of Mathematics and Statistics of the University of South Florida,the State Administration of Foreign Experts Affairs of China,the Natural Science Foundation of Shanghai(No.09ZR1410800)the National Natural Science Foundation of China(Nos.10971136,10831003,61072147 and 11071159)Chunhui Plan of the Ministry of Education of China.J.H.Meng and W.X.Ma/Adv.Appl.Math.Mech.,5(2013),pp.652-670669 References。
文摘We propose a class of non-semisimple matrix loop algebras consisting of 3×3 block matrices,and form zero curvature equations from the presented loop algebras to generate bi-integrable couplings.Applications are made for the AKNS soliton hierarchy and Hamiltonian structures of the resulting integrable couplings are constructed by using the associated variational identities.
文摘By introducing an invertible linear transform, a new Lie algebra G is obtained from the Lie algebra H. Making use of the compatibility conditions of the respective isospectral problems, a generalized NLS-MKdV hierarchy and a new integrable soliton hierarchy are achieved by using the trace identity or the variational identity. Then, two special non-semisimple Lie algebras ?and ?are explicitly conducted. As an application, the nonlinear continuous integrable couplings of the obtained integrable systems as well as their bi-Hamiltonian structures are established, respectively.
基金Supported in part by the Department of Mathematics and Statistics of University of South Floridathe State Administration of Foreign Experts Affairs of China+2 种基金the Natural Science Foundation of Shanghai under Grant No.09ZR1410800the Shanghai Leading Academic Discipline Project No.J50101the National Natural Science Foundation of China under Grant Nos.11271008,61072147,and11071159
文摘We propose a systematic approach for generating Hamiltonian tri-integrable couplings of soliton hierarchies. The resulting approach is based on semi-direct sums of matrix Lie algebras consisting of 4× 4 block matrix Lie algebras. We apply the approach to the AKNS soIRon hierarchy, and Hamiltonian structures of the obtained tri-integrable couplings are constructed by the variational identity.
