the two parameter quantum groups‎ ‎$u_{r,s}(mathfrak{g})$ associated to generalized kac-moody algebra‎ ‎and their equitable presentation

we construct a family of two parameter quantum grou-ps‎ ‎$u_{r,s}(mathfrak{g})$ associated with a generalized kac-moody‎ ‎algebra corresponding to symmetrizable admissible borcherds cartan‎ ‎matrix‎. ‎we also construct the $textbf{a}$-form $u_{textbf{a}}$ and‎ ‎the classical limit of $u_{r,s}(mathfrak{g})$‎. ‎furthermore‎, ‎we‎ ‎display the equitable presentation for a subalgebra‎ ‎$u_{r,s}^{b-}(mathfrak{g} )$ of $u_{r,s}(mathfrak{g})$ and show‎ ‎that this presentation has the attractive feature that all of its‎ ‎generators act semisimply on finite dimensional irreducible‎ ‎$u_{r,s}(mathfrak{g})$-modules associated with the kac-moody algebra‎.