CHAPTER 3 - CARTAN SUBALGEBRAS

Abstract

This chapter focuses on Cartan subalgebras. A Cartan subalgebra h of g is a maximal nilpotent subalgebra. Cartan subalgebras and Cartan decompositions of the classical Lie algebras are given as examples. It is shown that the two properties are common to Cartan subalgebras and Cartan decompositions of all semisimple Lie algebras. By Engel's theorem, g is a nilpotent Lie algebra. Every Lie algebra g contains a Cartan subalgebra. Let h be a nilpotent subalgebra of g that contains a regular element of g, then g°ad h is a Cartan subalgebra of g. If f is polynomial mapping from V to W and g is a polynomial mapping from W to U, then h = g o f is a polynomial mapping from V to U.