• ROOT MULTIPLICITIES FOR A CLASS OF QUASI AFFINE GENERALIZED KAC MOODY ALGEBRAS QAGGA2(1) OF RANK 4

Abstract

In this paper the concept of quasi affine generalized Kac Moody algebras (GKM) QAGGA₂⁽¹⁾ are defined; the general form of connected Dynkin diagrams associated with this particular class of QAGGA₂⁽¹⁾ are classified. Then, root multiplicities for some families of quasi affine generalized Kac Moody algebras are computed; First, we consider a general family QAGGA₂⁽¹⁾ of symmetrizable Generalized Generalized Cartan Matrices (GGCM) of quasi affine type, with one imaginary simple root,  of rank 4 which are obtained from the affine GCM A₂⁽¹⁾ of rank 3. The real simple  and imaginary simple roots for 3 different classes under this family QAGGA₂⁽¹⁾ are explicitly computed; Finally the root multiplicities of all roots of  GKM algebras  QAGGA₂⁽¹⁾ associated with these Borcherds Cartan matrices of order 4, which are obtained as extensions of the affine family A₂⁽¹⁾ are then determined;