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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.
ACKNOWLEDGMENT
The first author thanks the Department of Science and Technology, Government of India, New Delhi for the financial support for the Project No. DST/MS/050/96. The second author thanks University Grants Commission, Government of India, Southwestern Region, Bangalore for providing fellowship under Faculty Improvement Programme. Authors are thankful to the referee for some useful suggestions.