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 bik, cik, gik, hik (i ∈ I, k = 1,..., mi) of a Hopf algebra H to the quantized enveloping algebra Uq(G) of a Borcherds superalgebra G defined by a symmetrizable integral Borcherds–Cartan matrix A = (aij)i,j∈I. Therefore, we define an extended Hopf superalgebra H ⊗′ Uq(G). We also discuss the basis and the grouplike elements of H ⊗′ Uq(G).