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Component-trace identities for Hamiltonian structures. (English) Zbl 1192.37091
Summary: We show that on a particular class of semi-direct sums of matrix Lie algebras, component traces of the matrix product can produce bilinear forms which are non-degenerate, symmetric and invariant under the Lie product. The corresponding variational identities are called component-trace identities and provide tools in generating Hamiltonian structures of integrable couplings including the perturbation equations. An illustrative example of applying component-trace identities is given for the KdV hierarchy.
MSC:
37K05Hamiltonian structures, symmetries, variational principles, conservation laws
35Q53KdV-like (Korteweg-de Vries) equations
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37K30Relations of infinite-dimensional systems with algebraic structures