The extension of a quantized Borcherds superalgebra by a Hopf algebra

A two-parameter quantum group is obtained from the usual enveloping algebra by adding two commutative grouplike elements. In this paper, we generalize this procession further by adding commutative grouplike elements b ik , c ik , g ik , h ik (i . I, k = 1,..., m i ) of a Hopf algebra H to the quantized enveloping algebra U q (G) of a Borcherds superalgebra G defined by a symmetrizable integral Borcherds.Cartan matrix A = (a ij ) i,j.I . Therefore, we define an extended Hopf superalgebra H .. U q (G). We also discuss the basis and the grouplike elements of H .. U q (G).