THE KILLING FORMS AND DECOMPOSITION THEOREMS OF LIE SUPERTRIPLE SYSTEMS

In this article, the Killing form of a Lie supertriple system （LSTS） and that of its imbedding Lie superalgebra （LSA） are investigated, and a unique decomposition theorem for a quasiclassical LSTS with trivial center is established by means of the parallel decomposition theorem for a quasiclassical LSA.

The Killing Forms of Lie Triple Systems

For Lie triple systems in the characteristic zero setting, we obtain by means of the Killing forms two criterions for semisimplicity and for solvability respectively, and then investigate the relationship among the Killing forms of a real Lie triple system To, the complexification T of To, and the realification of T.

The extended trace identity and its application

The trace identity is extended to the general loop algebra. The Hamiltonian structures of the integrable sys- tems concerning vector spectral problems and the multi-component integrable hierarchy can be worked out by using the extended trace identity. As its application, we have obtained the Hamiltonian structures of the Yang hierarchy, the Korteweg-de-Vries （KdV） hierarchy, the multi-component Ablowitz-Kaup-Newell-Segur （M-AKNS） hierarchy, the multi-component Ablowitz-Kaup-Newell-Segur Kaup-Newell （M-AKNS-KN） hierarchy and a new multi-component integrable hierarchy separately.

Dynamics of some new solutions to the coupled DSW equations traveling horizontally on the seabed

The system of(1+1)-coupled Drinfel’d-Sokolov-Wilson equations describes the surface gravity waves travelling horizontally on the seabed.The objective of the present research is to construct a new variety of analytical solutions for the system.The invariants are derived with the aid of Killing form by using the optimal algebra classification via Lie symmetry approach.The invariant solutions involve time,space variables,and arbitrary constants.Imposing adequate constraints on arbitrary constants,solutions are represented graphically to make them more applicable in designing sea models.The behavior of solutions shows asymptotic,bell-shaped,bright and dark soliton,bright soliton,parabolic,bright and kink,kink,and periodic nature.The constructed results are novel as the reported results[26,28,29,30,33,38,42,49]can be deduced from the results derived in this study.The remaining solutions derived in this study,are absolutely different from the earlier findings.In this study,the physical character of analytical solutions of the system could aid coastal engineers in creating models of beaches and ports.

Some Results on Completly Restricted Lie Superalgebras

This paper is devoted to the study of completely restricted Lie superalgebras. We give some sufficient and necessary conditions for both completely restricted Lie superalgebras and strongly completely restricted Lie superalgebras. Some other important results on completely restricted Lie superalgebras are also obtained.

Note on Co-split Lie Algebras

It is proved that, any finite dimensional complex Lie algebra/~ = [Z：, ~：], hence, over a field of characteristic zero, any finite dimensional Lie algebra l： = [/2, ~：] possessing a basis with complex structure constants, should be a weak co-split Lie algebra. A class of non-semi-simple co-split Lie algebras is given.