Root Supermultiplicities of Some Borcherds Superalgebras

Abstract

In this article, we compute explicitly the supermultiplicities of the roots for the following Borcherds superalgebras associated with the colored Borcherds–Cartan matrices: (i) three different types of extended-hyperbolic Borcherds superalgebras; (ii) Borcherds superalgebras which are extensions by one simple imaginary root of finite type A 3, affine type , and hyperbolic type HA 1; (iii) Borcherds superalgebras with finite number of imaginary simple roots which are not mutually orthogonal but no real simple root; and (iv) Borcherds superalgebras with one real simple root and two nonorthogonal imaginary simple roots. The Kang and Kim superdimension formula is used for all the cases.