A subalgebra of Lie algebra A2 and its associated two types of loop algebras, as well as Hamiltonian structures of integrable hierarchy

In this paper, a subalgebra Ȧ2 of the Lie algebra A2 is constructed, which gives a corresponding loop algebra A¯2 by properly choosing the gradation of the basis elements. It follows that an isospectral problem is established and a new Liouville integrable Hamiltonian hierarchy is obtained. By making use of a matrix transformation, a subalgebra Ȧ2 of the Lie algebra A1 is presented, which possesses the same communicative operations of basis elements as those in Ȧ2. Again we expand the Lie algebra Ȧ1 into a high-dimensional loop algebra G̃, and a type of expanding integrable system of the hierarchy obtained above is worked out. Furthermore, Hamiltonian structures of hierarchy are presented by use of the quadratic form identity.