The two parameter quantum groups‎ ‎$U_{r,s}(mathfrak{g})$ associated to generalized Kac-Moody algebra‎ ‎and their equitable presentation

Abstract

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.