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).