The theory of partial differential operators is one of the most important branches of mathematics with several consequences in many other mathematical fields and with applications in other sciences. This project, which is of theoretical nature, intends to investigate the validity of the Fefferman-Phong, the Hörmander and the Melin inequalities for partial differential operators, and in general for pseudo-differential operators, on compact Lie groups, and apply them to the problem of solvability of degenerate partial differential operators. The analysis of partial differential operators requires the study of geometric quantities attached to the operators, in particular, the (total) symbol, the principal symbol and the subprincipal symbol. However, in the context of compact Lie groups, the principal symbol is globally well-defined but it is not the same for the other symbols mentioned above. Our goal is to define in a suitable way the other geometric quantities needed in the analysis of the problem and use them to obtain lower bounds for partial differential operators on compact Lie groups (i.e. the Fefferman-Phong, the Hörmander and the Melin inequalities). These lower bounds will be used to treat the problem of solvability of partial differential operators on compact Lie groups. We remark that the validity of these inequalities will yield the development of several results in the theory of partial differential equations on compact Lie groups, as, for instance, in the problems related to solvability, hypoellipticity, and well-posedness of the (weakly-hyperbolic) Cauchy problem.