Topological groups and Global properties[MatSciRep:63]

Abstract

These lecture notes are intended as an introductory course into the topological and global aspects of group theory. The lectures are kept simple in order to be of value to a physicist who is not very familiar with group theory. The presentation of the material is kept on a more intuitive than rigorous basis. Due to the choice of presentation, the lecture notes are of value to a greater number of physicists. Some work on these notes are done at Duke University. It comprises of eight Lectures in Group Theory. Concepts like Abstract groups, Subgroups, factor groups, Topological Spaces, Manifolds, Relations to the group aspects, Lie groups, Connected Lie Groups, Compact lie groups, Cartan subgroup, Cartan-Steifel diagrams, Weyl group, lattices, and properties of weights are discussed in these lectures. The first lecture deals with abstract groups and their algebraic properties while the second lecture introduces notion of the Topological spaces. The third lecture proves that subgroups of topological groups are again topological groups. Lie groups a particular type of Topological groups are discussed in the lecture 4 and the separability property of a topological space, the axiom of separability and Hausdorff Space are discussed here. A Hausdorff space (separable topological space) which is locally euclidean is called a topological manifold. This lecture concentrates particularly on analytic manifolds. Semi simple compact connected Lie groups are studied in detail along the the fifth lecture and finds the Cartan subgroups H, for the classical compact groups G. Sixth lecture concentrates on the embedding of these Cartan Subgroups, in the Topological space of the groups G, and determines the linear forms for the adjoint representation of SU(3). The next lecture aims to find these linear forms for the other classical groups, viz., (SO(2+1), SO(2), and Sp(2)). The eighth lecture deals with Weyl Groups, and it shows that the components of the roots are given by the coefficients of linear forms, and hence proves the statement, 'The roots just define the singular hyperplanes' - the linear forms.