J. Korean Math. Soc. 1998; 35(1): 127-134
Printed March 1, 1998
Copyright © The Korean Mathematical Society.
J. H. Keum and M. K. Lee
Konkuk University and Dongyang University
It is proved [6] that there exists a basis of $L^\Gamma$ (the space of meromorphic vector fields on a Riemann surface, holomorphic away from two fixed points) represented by the vector fields which have the expected zero or pole order at the two points. In this paper, we carry out the same task for the quadratic differentials. More precisely, we compute a basis of $Q^\Gamma$ (the space of meromorphic quadratic differentials on a Riemann surface, holomorphic away from two fixed points). This basis consists of the quadratic differentials which have the expected zero or pole order at the two points. Furthermore, we show that $Q^\Gamma$ has a Lie algebra structure which is induced from the Krichever-Novikov algebra $L^\Gamma$.
Keywords: Riemann surfaces, quadratic differentials, Krichever-Novikov algebra
MSC numbers: 14H55
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