On the construction of semisimple Lie algebras and Chevalley groups

Source abstract

Let g \mathfrak {g} be a semisimple complex Lie algebra. Recently, Lusztig simplified the traditional construction of the corresponding Chevalley groups (of adjoint type) using the “canonical basis” of the adjoint representation of g \mathfrak {g} . Here, we present a variation of this idea which leads to a new, and quite elementary, construction of g \mathfrak {g} itself from its root system. An additional feature of this set-up is that it also gives rise to explicit Chevalley bases of g \mathfrak {g} .