THE TWO PARAMETER QUANTUM GROUPS U r,s(g) ASSOCIATED TO GENERALIZED KAC-MOODY ALGEBRA AND THEIR EQUITABLE PRESENTATION

We construct a family of two parameter quantum groups U r,s(g) associated with a generalized Kac-Moody algebra corresponding to symmetrizable admissible Borcherds Cartan matrix. We also construct the A-form UA and the classical limit of Ur,s(g). Furthermore, we display the equitable presentation for a subalgebra U b- r,s (g) of U r,s(g) and show that this presentation has the attractive feature that all of its generators act semisimply on finite dimensional irreducible U r,s(g)-modules associated with the Kac-Moody algebra.