  In this paper we give several global characterisations of the Hormander class
of pseudo-differential operators on compact Lie groups. The result is applied
to give criteria for the ellipticity and the global hypoellipticity of
pseudo-differential operators in terms of their matrix-valued full symbols.
Several examples of the first and second order globally hypoelliptic
differential operators are given. Where the global hypoelliptiticy fails, one
can construct explicit examples based on the analysis of the global symbols.
