Hamiltonian and super-Hamiltonian extensions related to Broer-Kaup-Kupershmidt system. (English) Zbl 1185.81096
Summary: Based on the Lie algebra \(A_1\), the integrable Broer-Kaup-Kupershmidt (BKK) system is revisited. The bi-Hamiltonian structure is constructed by the trace identity. Two extensions of the Lie algebra \(A_1\) are considered, i.e., the non-semi-simple Lie algebra of \(4\times 4\) matrix and the super-Lie algebra of \(3\times 3\) matrix, from which two hierarchies of soliton equations related to BKK system are given. With the aid of the generalized trace identity and the super-trace identity, the Hamiltonian and super-Hamiltonian structures of the resulting systems are constructed.
MSC:
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 17B80 | Applications of Lie algebras and superalgebras to integrable systems |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
Keywords:
non-semi-simple Lie algebras; super-Lie algebras; generalized trace identity; super-trace identity; super-Hamiltonian structureReferences:
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