Properties of the Derived Algebra of a Semisimple Lie Algebra

Introduction

A semisimple Lie algebra is a Lie algebra that is a direct sum of simple Lie algebras. Over an algebraically closed field, such as the complex numbers, semisimple Lie algebras are characterized by their decomposition into simple components. The derived algebra of a Lie algebra, denoted $[g, g]$, is the subspace generated by all commutators of elements in $g$. For semisimple Lie algebras, the derived algebra exhibits unique properties that distinguish it from non-semisimple algebras. This article explores the structure, properties, and implications of the derived algebra of a semisimple Lie algebra, emphasizing its relationship to the Lie algebra itself and its role in representation theory.

Definition and Properties

The derived algebra of a Lie algebra $g$ is defined as:
$$ [ g, g ] = { x y - y x \mid x, y \in g }. $$
For a semisimple Lie algebra $g$, the derived algebra coincides with the entire algebra $g$ itself. This is because the derived algebra is the subspace generated by all commutators, and in a semisimple Lie algebra, the commutators of elements from different simple components can be used to generate the entire algebra. Specifically, if $g = \mathfrak{g}_1 \oplus \mathfrak{g}_2 \oplus \cdots \oplus \mathfrak{g}_k$, where each $\mathfrak{g}_i$ is a simple Lie algebra, then the derived algebra of $g$ is:
$$ [ g, g ] = \mathfrak{g}_1 \oplus \mathfrak{g}_2 \oplus \cdots \oplus \mathfrak{g}_k = g. $$
This result is a direct consequence of the fact that the derived algebra of a semisimple Lie algebra is equal to the Lie algebra itself. Thus, the derived algebra of a semisimple Lie algebra is trivial in the sense