A new loop algebra and its expanded loop algebras, as well as their applications

A Lie algebra, whose bases are the forms of M×3M×3 matrices, is defined. It follows that two types loop algebras are constructed, whose commutative operations are equivalent to the known ones proposed before. By making use of Tu scheme, the multicomponent KN hierarchy and its integrable coupling system, as well as a generalized multicomponent AKNS integrable hierarchy with five potential functions are obtained, respectively. We find that the procedure presented in this paper is rather simple and straightforward, which can be used generally.