Solvable and Semisimple Algebras - Main Theorem

The fundamental theorems of structure theory provide a complete understanding of how Lie algebras can be decomposed and classified. These results form the theoretical foundation for all of Lie theory.

Theorem

Levi Decomposition Theorem

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field of characteristic zero. Then $\mathfrak{g}$ can be written as a semidirect product: $\mathfrak{g} = \mathfrak{s} \ltimes \text{Rad}(\mathfrak{g})$ where $\mathfrak{s}$ is a semisimple Lie algebra (called a Levi subalgebra) and $\text{Rad}(\mathfrak{g})$ is the radical. Moreover, any two Levi subalgebras are conjugate by an automorphism of the form $\exp(\text{ad}_X)$ for $X \in \text{Rad}(\mathfrak{g})$ .

The Levi decomposition reduces the study of arbitrary Lie algebras to understanding semisimple algebras and solvable algebras separately, plus how they fit together in a semidirect product.

Theorem

Cartan's Criterion for Semisimplicity

A Lie algebra $\mathfrak{g}$ over a field of characteristic zero is semisimple if and only if its Killing form $\kappa(X, Y) = \text{tr}(\text{ad}_X \circ \text{ad}_Y)$ is non-degenerate.

This provides a computational test for semisimplicity that avoids finding all ideals. For matrix algebras, it reduces to checking the rank of a matrix formed from traces.

Remark

The Killing form is functorial: for any ideal $\mathfrak{i} \subseteq \mathfrak{g}$ , the restriction of the Killing form of $\mathfrak{g}$ to $\mathfrak{i}$ equals the Killing form of $\mathfrak{i}$ . This property is crucial in proving Cartan's criterion.

Theorem

Weyl's Theorem on Complete Reducibility

Let $\mathfrak{g}$ be a semisimple Lie algebra over a field of characteristic zero. Then every finite-dimensional representation of $\mathfrak{g}$ is completely reducible.

Weyl's theorem is the cornerstone of representation theory for semisimple Lie algebras. It implies that to understand all representations, one need only classify irreducible representations and count multiplicities.

Theorem

Malcev's Theorem

Two finite-dimensional Lie algebras over an algebraically closed field of characteristic zero are isomorphic if and only if their universal enveloping algebras are isomorphic as associative algebras.

This establishes that the Lie algebra structure is completely encoded in its universal enveloping algebra, providing an alternative algebraic perspective.

Theorem

Classification of Simple Lie Algebras

Over an algebraically closed field of characteristic zero, the simple Lie algebras are:

The classical series: $A_n$ ( $\mathfrak{sl}_{n+1}$ ), $B_n$ ( $\mathfrak{so}_{2n+1}$ ), $C_n$ ( $\mathfrak{sp}_{2n}$ ), $D_n$ ( $\mathfrak{so}_{2n}$ )
The exceptional algebras: $G_2, F_4, E_6, E_7, E_8$

This classification is achieved via root systems and Dynkin diagrams, which we study in detail later.

Remark

The classification theorem is one of the pinnacles of 19th and 20th century mathematics. It shows that despite the apparent complexity of Lie algebras, there is a beautiful finite list of building blocks from which all semisimple Lie algebras are constructed.

Casimir element: For semisimple $\mathfrak{g}$ with basis $\{X_i\}$ dual to $\{X^i\}$ under the Killing form, the element $C = \sum_i X_i X^i \in U(\mathfrak{g})$ is central and acts by scalars on irreducible representations, providing powerful invariants.