A fundamental representation of quantum generalized Kac-Moody algebras with one imaginary simple root

Abstract

We consider the Borcherds-Cartan matrix obtained from a symmetrizable generalized Cartan matrix by adding one imaginary simple root. We extend the result of Gebert and Teschner (Lett. Math. Phys., 1994, 31: 327-334) to the quantum case. Moreover, we give a connection between the irreducible dominant representations of quantum Kac-Moody algebras and those of quantum generalized Kac-Moody algebras. As the result, a large class of irreducible dominant representations of quantum generalized Kac-Moody algebras were obtained from representations of quantum Kac-Moody algebras through tensor algebras.