  A new approach to analyze the properties of the energy-momentum tensor $T(z)$
of conformal field theories on generic Riemann surfaces (RS) is proposed.
$T(z)$ is decomposed into $N$ components with different monodromy properties,
where $N$ is the number of branches in the realization of RS as branch covering
over the complex sphere. This decomposition gives rise to new infinite
dimensional Lie algebra which can be viewed as a generalization of Virasoro
algebra containing information about the global properties of the underlying
RS. In the simplest case of hyperelliptic curves the structure of the algebra
is calculated in two ways and its central extension is explicitly given. The
algebra possess an interesting symmetry with a clear interpretation in the
framework of the radial quantization of CFT's with multivalued fields on the
complex sphere.
