An approach for constructing loop algebra via exterior algebra and its applications

With the help of some properties of exterior algebra defined by us, a general approach for constructing multi-component matrix loop algebra is proposed. By making use of the approach, a new 3M loop algebra X is constructed. This algebra can be easily reduced to the existing multi-component loop algebra. Another an new extended loop algebra Y is also presented. As their applicable examples, a generalized multi-component AKNS hierarchy with arbitrary smooth functions and a generalized multi-component KN hierarchy are worked out. As a reduction cases of the first hierarchy, the standard multi-component heat-conduction equation and a coupled generalized multi-component Burgers equation are given. The approach presented in the paper can be used generally.