Abstract
In this paper, we find out all imaginary roots and special imaginary roots of generalized Kac-Moody algebras (other than Kac-Moody algebras) whose symmetrizable generalized generalized Cartan matrices (of order ) are extensions of Cartan matrices (of order ) of finite, affine and hyperbolic types. We notice that in the classes of generalized Kac-Moody algebras (other than Kac-Moody algebras) whose symmetrizable generalized generalized Cartan matrices (of order are extensions of Cartan matrices (of order ) of finite type, some algebras possess special imaginary roots whereas some others do not. Also we find that the classes of generalized Kac-Moody algebras (other than Kac-Moody algebras) whose symmetrizable generalized Cartan matrices (of order ) are extensions of Cartan matrices (of order ) of affine and hyperbolic types, do not possess special imaginary roots whereas the underlying hyperbolic Kac-Moody algebras possess special imaginary roots.
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