ABSTRACT
One of the main achievements in the theory of algebraic groups is the classification of the semisimple ones, due to the work of Chevalley. This classification has many intriguing aspects, one being its independence on the characteristic of the ground field: the semisimple algebraic groups over a given algebraically closed field k are, up to isomorphism, parametrized by the semisimple root data. This chapter utilizes Steinberg’s approach to the construction of the semisimple algebraic groups. It describes an algorithm that, given a highest weight representation of a semisimple algebraic group and a matrix lying in the image of that representation, gives a preimage of that matrix expressed as a normal word. The chapter explains the Chevalley groups constructed from a representation of a semisimple Lie algebra and a field. It shows in strengthened form for the Chevalley groups.
