Lie Algebras - Main Theorem

The structure theory of Lie algebras culminates in several fundamental theorems that characterize semisimple and solvable Lie algebras. These results form the foundation for the classification of Lie algebras and their representations.

Theorem

Cartan's Criterion for Semisimplicity

A finite-dimensional 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.

The Killing form is a symmetric bilinear form on $\mathfrak{g}$ that is invariant: $\kappa([X, Y], Z) = \kappa(X, [Y, Z])$ . This invariance follows from the cyclic property of the trace and makes the Killing form a fundamental tool in Lie theory.

Remark

For simple Lie algebras, the Killing form is not just non-degenerate but actually provides a canonical inner product (up to scaling). This inner product is used to define the root system and construct highest weight representations.

Theorem

Weyl's Complete Reducibility Theorem

Every finite-dimensional representation of a semisimple Lie algebra over a field of characteristic zero is completely reducible, i.e., it decomposes as a direct sum of irreducible representations.

This theorem is the Lie algebra analogue of Maschke's theorem for finite groups. It drastically simplifies the representation theory of semisimple Lie algebras: to understand all representations, one need only classify the irreducible ones.

Example

For $\mathfrak{sl}_2(\mathbb{C})$ , the irreducible representations $V_n$ are indexed by non-negative integers $n$ (corresponding to dimension $n+1$ ). Every representation of $\mathfrak{sl}_2(\mathbb{C})$ decomposes uniquely as: $V = \bigoplus_{n=0}^\infty V_n^{\oplus m_n}$ for some multiplicities $m_n \geq 0$ .

Theorem

Lie's Theorem

Let $\mathfrak{g}$ be a solvable Lie algebra over an algebraically closed field of characteristic zero, and let $\rho: \mathfrak{g} \to \mathfrak{gl}(V)$ be a finite-dimensional representation. Then there exists a basis of $V$ in which all operators $\rho(X)$ for $X \in \mathfrak{g}$ are upper triangular.

Lie's theorem shows that solvable Lie algebras always have common eigenvectors (or more precisely, weight vectors). This result is crucial for studying the structure of general Lie algebras via their solvable subalgebras.

Theorem

Engel's Theorem

If $\mathfrak{g}$ is a Lie algebra such that $\text{ad}_X$ is nilpotent for every $X \in \mathfrak{g}$ , then $\mathfrak{g}$ is nilpotent. Equivalently, for a finite-dimensional representation where every $\rho(X)$ is nilpotent, there exists a basis where all $\rho(X)$ are strictly upper triangular.

The combination of Lie's and Engel's theorems provides powerful tools for understanding the structure of non-semisimple Lie algebras. Together with the Levi decomposition (splitting a Lie algebra into semisimple and solvable parts), these theorems reduce many questions to the semisimple case, which is then handled by root system theory and the classification theorem.