Construct Weak Hopf Algebras By Using Borcherds Matrix

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We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra G by adding a new generator J satisfying Jᵐ=J for some integer m. We denote this algebra by wUqτ(G). 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 usually quantum envelope algebra Uq(G) of a generalized Kac-Moody algebra G.