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

摘要

In this paper, a subalgebra A ˙n2 of the Lie algebra A2 is constructed, which gives ancorresponding loop algebra A¯n2 by properly choosing the gradation of the basisnelements. It follows that an isospectral problem is established and a new Liouvillenintegrable Hamiltonian hierarchy is obtained. By making use of a matrix transformation,na subalgebra A ˙n2 of the Lie algebra A1 is presented, which possesses thensame communicative operations of basis elements as those in A ˙ 2. Again we expandnthe Lie algebraA ˙n1 into a high-dimensional loop algebraG ˜ , and a type of expandingnintegrable system of the hierarchy obtained above is worked out. Furthermore,nHamiltonian structures of hierarchy are presented by use of the quadratic formnidentity.