A Note on Purely Imaginary Roots of Generalized Kac–Moody Algebras

Abstract

In this paper, we generalize the concept of purely imaginary roots of Kac–Moody algebras to generalized Kac–Moody algebras. Also we give a complete classification of those generalized Kac–Moody algebras with the purely imaginary property. We also define a new class of indefinite non-hyperbolic generalized Kac–Moody algebras called extended hyperbolic generalized Kac–Moody algebras and find that it does not always possess the purely imaginary property whereas the extended hyperbolic Kac–Moody algebras possess the purely imaginary property.

Key Words:

• Generalized Kac–Moody algebras
• Root systems

Acknowledgments

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 suggesting minor editing improvements.