## 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 [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 infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 17B80 Applications of Lie algebras and superalgebras to integrable systems

Zbl 0678.70015
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