ON THE FUNDAMENTAL REPRESENTATION OF BORCHERDS ALGEBRAS WITH ONE IMAGINARY SIMPLE ROOT

Borcherds algebras represent a new class of Lie algebras which have al most all the properties that ordinary Kac-Moody algebras have, but the only major difference is that these generalized Kac-Moody algebras ar e allowed to have imaginary simple roots. The simplest nontrivial exam ples one can think of are those where one adds 'by hand' one imaginary simple root to an ordinary Kac-Moody algebra. We study the fundamenta l representation of this class of examples and prove that an irreducib le module is given by the full tenser algebra over some integrable hig hest weight module of the underlying Kac-Moody algebra. We also commen t on possible realizations of these Lie algebras in physics as symmetr y algebras in quantum field theory.