A combinatorial description of the good Z-gradings of the symplectic Lie algebra

In this thesis we investigate the concept of good Z-grading of a finite dimensional semisimple Lie algebra over an algebraically closed field of characteristic 0. Throughout the thesis we make use of the theorem of Jacobson Morozov, a fundamental result in the Lie theory. First of all we give a fundamental example of good grading, the Dynkin one. Afterwards we study more in general some properties of the good Z-gradings of a Lie algebra. Finally we give a complete description of the good Z-gradings of the symplectic Lie algebra.