In this talk, I will tell you about the Borel-de-Sibenthal theorem which gives the classification of all maximal closed subroot systems of finite crystallographic root systems. I will start my talk by introducing the notion of finite root systems and it’s closed subroot systems.

The concept of root system is very fundamental in the theory of Lie groups and Lie algebras. Especially they play a vital role in the classification of finite dimensional semi-simple Lie algebras. Closed subroot systems of finite root systems naturally appear in the Borel-de-Sibenthal theory which describes the closed connected subgroups of a compact Lie group that have maximal rank. The classification of closed subroot systems is essential in the classification of semi-simple subalgebras of semi-simple Lie algebras.

Through out this talk, we will try to stay within the theory of root systems and reflection groups. No knowledge of Lie algebras or Lie groups will be assumed. If time permits I will discuss about my joint work with R. Venkatesh which gives explicit descriptions of the maximal closed subroot systems of affine root systems.