Title: Construct weak Hopf algebras by using Borcherds matrix
Abstract: We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra by adding a new generator J satisfying J m = J for some integer m. We denote this algebra by wU ( ). This algebra is a weak Hopf algebra if and only if m = 2, 3. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usual quantum envelope algebra U q ( ) of a generalized Kac-Moody algebra .