Definition of solvable Lie algebras, derived series, and determination

Lie algebras are fundamental structures in mathematics, serving as a natural generalization of vector spaces and playing a pivotal role in the study of Lie groups, differential geometry, and representation theory. A Lie algebra is a vector space equipped with a bilinear operation called the Lie bracket, which satisfies certain axioms, including antisymmetry and the Jacobi identity. The derived series, a sequence of subalgebras generated by successive Lie brackets, provides a powerful tool to analyze the structure of Lie algebras. This article explores the definition of solvable Lie algebras, the properties of the derived series, and the method for determining solvability.

Definition of Lie Algebras

A Lie algebra over a field $ \mathbb{F} $ is a vector space $ \mathfrak{g} $ over $ \mathbb{F} $ endowed with a bilinear operation $ [\cdot, \cdot] : \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g} $ satisfying the following properties:

Antisymmetry: $ [x, x] = 0 $ for all $ x \in \mathfrak{g} $.
Jacobi Identity: $ [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 $ for all $ x, y, z \in \mathfrak{g} $.

These structures are central to the study of Lie groups and their tangent spaces. The derived series, a sequence of subalgebras $ { \mathfrak{g}^i } $, is defined recursively as $ \mathfrak{g}^0 = \mathfrak{g} $ and $ \mathfrak{g}^{i+1} = [\mathfrak{g}^i, \mathfrak{g}^i] $. The series terminates when $ \mathfrak{g}^{i+1} = 0 $, at which point the Lie algebra is said to be solvable.

Derived Series

The derived series of a Lie algebra $ \mathfrak{g} $ is a sequence of subalgebras that progressively captures the "non-commutative" structure of $ \mathfrak{g} $. The first term, $ \mathfrak{g}^0 $, is the algebra itself, while $ \mathfrak{g}^1 = [\mathfrak{g}, \mathfrak{g}] $ represents the commutator subalgebra. Subsequent terms are obtained by applying the Lie bracket iteratively:
$$ \mathfrak{g}^{i+1} = [\mathfrak{g}^i, \mathfrak{g