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A multi-component matrix loop algebra and a unified expression of the multi-component AKNS hierarchy and the multi-component BPT hierarchy. (English) Zbl 1222.37072
Summary: A set of multi-component matrix Lie algebras is constructed, which is devoted to obtaining a new loop algebra $\tilde A_{M-1}$. It follows that an isospectral problem is established. By making use of the Tu scheme [{\it G. Tu}, J. Math. Phys. 30, No. 2, 330--338 (1989; Zbl 0678.70015)], a Liouville integrable multi-component hierarchy of soliton equations is generated, which possesses the bi-Hamiltonian structures. As its reduction cases, the multi-component AKNS hierarchy and the formalism of the multi-component BPT hierarchy are given, respectively.

##### MSC:
 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 17B80 Applications of Lie algebras to integrable systems
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##### References:
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