Pages Lie Algebras II: Structure and Classification Attributions

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This course develops the structural theory and classification of finite-dimensional Lie algebras over the complex numbers, with a focus on the semisimple case. After the introductory chapter, it studies how the Killing form detects semisimplicity and how Cartan subalgebras organize the internal geometry of a Lie algebra. From there, the course turns to the root space decomposition, which breaks a semisimple Lie algebra into a Cartan subalgebra together with weight spaces that encode its essential structure.

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The central theme is that semisimple Lie algebras can be understood through their roots. The course first builds the abstract theory of root systems, then introduces bases, positive roots, and Weyl chambers to impose combinatorial structure on the root data. Weyl groups, Cartan matrices, and Dynkin diagrams then provide the symmetry and encoding needed to pass from linear-algebraic information to a finite classification problem.

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The later chapters complete that classification. They show how finite root systems are classified, how a Lie algebra can be reconstructed from its root data, and why the resulting Lie algebra is determined uniquely up to isomorphism. In this way, the course moves from intrinsic structural tools to a complete and explicit description of semisimple Lie algebras, culminating in the classification theorem.

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# Introduction

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This course develops the classification of finite-dimensional semisimple Lie algebras over the complex numbers. The main problem is to turn a noncommutative algebra into intrinsic data that survives change of basis: first the Killing form and Cartan subalgebras, then root spaces, root systems, Weyl groups, Cartan matrices, and Dynkin diagrams.

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The path is deliberately circular. We begin with a semisimple Lie algebra and extract a finite root system from its adjoint action. We then classify the possible root systems by their diagrams, and finally reconstruct the Lie algebra from the resulting Cartan matrix by generators and relations. The point of the course is that this circle closes: no hidden continuous parameter remains once the finite-type Dynkin diagram is known.

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The notes assume the usual first-course material on ideals, solvable and nilpotent Lie algebras, Engel's theorem, Lie's theorem, the Killing form, and Cartan's criteria. Those facts are used as tools rather than reproved as a separate preliminary chapter. Chapter 1 begins the classification argument by isolating semisimplicity and explaining why the Killing form is the right bilinear invariant for the rest of the course.

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Throughout these notes the standing field is $\mathbb C$. When formulas use the letter $k$, read $k=\mathbb C$ unless a local example explicitly says otherwise. This keeps the quoted theorem cards, which are mostly stated in the classical complex form, aligned with the surrounding course text.

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# 1. Semisimple Lie Algebras and the Killing Form

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This opening chapter fixes the structural language used throughout the classification course. The central question is how much of a finite-dimensional Lie algebra is forced by the absence of solvable ideals, and why the Killing form is the correct bilinear form for detecting that condition. We work over $k=\mathbb C$, and all Lie algebras in this chapter are finite-dimensional over $k$ unless stated otherwise.

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## Structural Recap: Radicals, Semisimplicity, and the Adjoint Action

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What is the obstruction to treating a Lie algebra as a direct sum of indecomposable non-abelian pieces? The first obstruction is the solvable radical: it collects all solvable ideal behaviour into a single canonical ideal. Once that ideal vanishes, the Lie algebra becomes rigid enough for the Killing form and the adjoint action to control its structure.

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[definition: Solvable Radical] Let $L$ be a Lie algebra. The solvable radical of $L$, denoted $\operatorname{rad} L$, is the largest solvable ideal of $L$. [/definition]

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The existence of $\operatorname{rad} L$ follows because the sum of two solvable ideals is solvable. Thus the sum of all solvable ideals is again a solvable ideal, and it contains every other solvable ideal.

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With the radical available as a canonical obstruction, the next step is to name the case where that obstruction has vanished. This is the structural setting in which the Killing form will become nondegenerate.

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[definition: Semisimple Lie Algebra] A Lie algebra $L$ is semisimple if $\operatorname{rad} L = 0$. [/definition]

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This definition rules out abelian ideals, since every abelian ideal is solvable. It also rules out nonzero centres, because $Z(L)$ is an abelian ideal.

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Semisimplicity removes solvable pieces, but it does not require the ideal structure to be indivisible. To describe the building blocks of semisimple algebras, we next isolate the algebras with no proper nontrivial ideals at all.

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[definition: Simple Lie Algebra] A Lie algebra $L$ is simple if $L$ is non-abelian and its only ideals are $0$ and $L$. [/definition]

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Simple Lie algebras are the irreducible objects for the ideal structure, but semisimple Lie algebras need not be simple. The point of this chapter is that semisimple Lie algebras are still built from simple ideals in a controlled direct-sum way.

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