IJPAM: Volume 38, No. 4 (2007)

SPECIAL IMAGINARY ROOTS OF
BKM LIE SUPER ALGEBRAS

N. Sthanumoorthy$^1$, P.L. Lilly$^2$, A. Nazeer Basha$^3$
$^{1,3}$The Ramanujan Institute of Advanced Study in Mathematics
University of Madras
Chennai, 600 005, INDIA
$^1$e-mail: sthanun@yahoo.com
$^2$Department of Mathematics
St. Joseph's College
Irinjalakuda, 680 121, INDIA
e-mail: lillypl@rediffmail.com


Abstract.In this paper, we find out all imaginary roots and special imaginary roots of BKM Lie super algebras (Borcherds Kac-Moody Lie super algebras), other than BKM algebras, whose symmetrizable BKM super matrices (of order $\leq$ 4) are extensions of KM matrices (of order $\leq$ 3) of finite, affine and hyperbolic types. We notice that all BKM super algebras, whose BKM super matrices are the extensions of KM matrices of finite-non-simply laced types, have special imaginary roots, but from among the BKM super algebras whose BKM super matrices are extensions of KM matrices of simply laced finite type, some algebras possess special imaginary roots whereas some others do not. Also we find that all BKM super algebras whose BKM super matrices are extensions of KM matrices of affine and hyperbolic types do not possess special imaginary roots whereas the underlying hyperbolic KM algebras possess special imaginary roots. Moreover for BKM super algebras, we found that only the alien-type imaginary roots are special imaginary roots whereas the domestic-type imaginary roots are not special imaginary roots. We also notice that all alien-type imaginary roots are even imaginary roots.

Received: June 15, 2007

AMS Subject Classification: 17B65, 17B67

Key Words and Phrases: BKM Lie super algebras, root systems, special imaginary roots

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 38
Issue: 4