Maximal Torus in a Lie Group and its Weyl Group

Kumar, Neeraj (2017) Maximal Torus in a Lie Group and its Weyl Group. Masters thesis, Indian Institute of Science Education and Research, Kolkata.

PDF (MS dissertation of Neeraj Kumar (12MS043)) MS_Thesis_by_Neeraj_Kumar.pdf - Submitted Version Restricted to Repository staff only Download (303kB)

Abstract

This thesis involves the study of the maximal torus and the Weyl group in compact connected Lie groups. The study is based on the text Representations of Compact Lie Groups by Brocker and Dieck. The conjugation theorem (due to Cartan) is the main result in the first part of the study, which deals with the fact that the maximal tori in a Lie group are conjugate to one other and that these tori cover the entire group. This conjugacy induces an isomorphism of tori and Weyl groups, hence enabling us to talk about the Weyl group of the Lie group. Afterwards, some useful consequences are procured through this theorem in the next section, which aids the study of the action of Weyl group on the maximal torus. Finally, we look at some explicit examples of compact connected Lie groups through the case of compact classical groups and we compute the maximal torus and the Weyl group for these cases.

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