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

Document Type: Research Paper

Authors

1 Zhejiang University of Science and Technology, China

2 Zhejiang International Studies University, China

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.