<p>This video explores the simple roots introducing, in particular, key concepts as the Cartan matrix (which encodes the information about the relative angles of the simple roots) of fundamental weights (which are at the basis of the representation theory). It is part of a complete course consisting of 15 videos where there will be covered in details (together with many instructive examples) the basic properties of Group Theory of finite and continuous groups (such as the permutation group and the SU(2) group), the representation theory, the structure theory of roots and weights and the classification of compact Lie group. Symmetry is one of the most pervasive and powerful concepts in mathematics and its language is given by Group Theory, that is a quite universal discipline and finds vast applications in many other subjects, such as many branches of physics. Examples and exercises presented in this lesson will range from number theory to simple physical systems. The ideal viewers areundergraduate and graduate university students of scientific disciplines, postdoctoral fellows and, in general, scientific scholars eager to learn or deepen their knowledge on Group Theory.</p>