sl₄ℂ and sl_nℂ

Abstract

In this lecture, we will illustrate the general paradigm of the previous lecture by applying it to the Lie algebras$\mathfrak{s}{\mathfrak{l}_n}\mathbb{C}$ this is typical of the analyses of specific Lie algebras carried out in this Part. We start in §15.1 by describing the Cartan subalgebra, roots, root spaces, etc., for$\mathfrak{s}{\mathfrak{l}_n}\mathbb{C}$in general. We then give in §15.2 a detailed account of the representations of$\mathfrak{s}{\mathfrak{l}_n}\mathbb{C}$, which generalizes directly to$\mathfrak{s}{\mathfrak{l}_n}\mathbb{C}$in particular, we deduce the existence part of Theorem 14.18 for $\mathfrak{s}{{\mathfrak{l}}_{n}}\mathbb{C}$.