This course introduces the basic structure theory of Lie algebras, which are algebraic objects designed to encode infinitesimal symmetry. The opening chapters develop the definition and first examples, then move to the standard constructions that organize the subject: ideals, quotients, subalgebras, and representations. From there, the course shifts from examples and formalism to the behavior of Lie algebras acting on vector spaces, where module-theoretic ideas become a central tool.

A major theme is the distinction between solvable, nilpotent, and semisimple Lie algebras, and the way these classes are detected by structural criteria. The middle chapters build the machinery needed for this classification, culminating in Engel’s theorem, Lie’s theorem, and triangularization results. The final chapters study the Killing form and Cartan’s criterion, then use them to develop the theory of semisimple Lie algebras and prove Weyl’s theorem on complete reducibility. Each chapter adds a layer of structure that the next chapter uses, so the course moves steadily from foundational definitions to the core structural theorems of the subject.

# Introduction

This opening chapter sets the scope and conventions for **Lie Algebras I: Foundations**. The course studies finite-dimensional Lie algebras over a field, beginning with the bracket operation and moving toward the structural tests that separate solvable, nilpotent, and semisimple behaviour. The guiding theme is that many internal properties of a Lie algebra are best detected through its representations, especially the adjoint representation.

The notes assume linear algebra, abstract algebra, basic module theory, elementary field theory, and familiarity with [Jordan normal form](/page/Jordan%20Normal%20Form). Later chapters use these prerequisites without reintroducing them, but this chapter fixes the course vocabulary and explains why the first structural theorems take the form they do.

## What Is the Course Trying to Classify?

The central problem is to understand when two non-abelian algebraic systems built from a bilinear bracket should be regarded as structurally similar. Unlike groups or associative algebras, Lie algebras are governed by infinitesimal commutation: the bracket measures failure of two infinitesimal operations to commute. This makes Lie algebras both algebraic objects in their own right and linear shadows of Lie groups.

The course does not attempt the full classification of semisimple Lie algebras. Instead, it builds the foundational tools needed before classification can begin: ideals, quotient Lie algebras, solvability, nilpotence, representations, trace criteria, and complete reducibility.

[definition: Lie Algebra Course Object] A Lie algebra in this course is a finite-dimensional [vector space](/page/Vector%20Space) $\mathfrak g$ over a field $F$ equipped with a bilinear map \begin{align*} [-,-] : \mathfrak g \times \mathfrak g \to \mathfrak g \end{align*} satisfying skew-symmetry and the Jacobi identity. [/definition]

The definition is deliberately spare. Most of the course asks how much structure is forced by these two identities once finite-dimensionality is imposed.

[example: Commutators As The First Source] Let $A$ be an associative algebra over $F$, and define \begin{align*} [x,y]=xy-yx \end{align*} for $x,y\in A$. We show that this bracket is bilinear, skew-symmetric, and satisfies the Jacobi identity. For $a,b\in F$ and $x_1,x_2,y\in A$, distributivity and compatibility of scalar multiplication with multiplication in the associative algebra give \begin{align*} [ax_1+bx_2,y] &=(ax_1+bx_2)y-y(ax_1+bx_2)\\ &=a x_1y+b x_2y-a yx_1-b yx_2\\ &=a(x_1y-yx_1)+b(x_2y-yx_2)\\ &=a[x_1,y]+b[x_2,y]. \end{align*} The same calculation in the second variable gives \begin{align*} [x,ay_1+by_2] &=x(ay_1+by_2)-(ay_1+by_2)x\\ &=a xy_1+b xy_2-a y_1x-b y_2x\\ &=a[x,y_1]+b[x,y_2], \end{align*} so the bracket is bilinear. Skew-symmetry follows from \begin{align*} [y,x]=yx-xy=-(xy-yx)=-[x,y]. \end{align*} It remains to verify the Jacobi identity. Using associativity to remove parentheses in products of three elements, we compute \begin{align*} [x,[y,z]] &=x(yz-zy)-(yz-zy)x\\ &=xyz-xzy-yzx+zyx,\\ [y,[z,x]] &=y(zx-xz)-(zx-xz)y\\ &=yzx-yxz-zxy+xzy,\\ [z,[x,y]] &=z(xy-yx)-(xy-yx)z\\ &=zxy-zyx-xyz+yxz. \end{align*} Adding these three displayed identities term by term gives \begin{align*} [x,[y,z]]+[y,[z,x]]+[z,[x,y]] &=(xyz-xzy-yzx+zyx)\\ &\quad +(yzx-yxz-zxy+xzy)\\ &\quad +(zxy-zyx-xyz+yxz)\\ &=0, \end{align*} because $xyz$ cancels with $-xyz$, $xzy$ with $-xzy$, $yzx$ with $-yzx$, $zyx$ with $-zyx$, $yxz$ with $-yxz$, and $zxy$ with $-zxy$. Thus the commutator bracket makes every associative algebra into a Lie algebra. [/example]

This example explains why matrix Lie algebras dominate the first lectures. The operation $XY-YX$ is concrete enough for computation, but it already contains the identities that govern abstract Lie algebras.

## Why Ideals and Quotients Come First

The first structural question is how to break a Lie algebra into smaller pieces without destroying the bracket. Subalgebras are closed under the bracket, but ideals are the correct analogue of normal subgroups because they are exactly the subspaces that support quotient Lie algebras.

[definition: Ideal] Let $\mathfrak g$ be a Lie algebra over $F$. An ideal of $\mathfrak g$ is a vector subspace $\mathfrak i\subset\mathfrak g$ such that \begin{align*} [x,y]\in\mathfrak i \end{align*} for all $x\in\mathfrak g$ and $y\in\mathfrak i$. [/definition]

Ideals measure internal compatibility with all bracket operations from $\mathfrak g$. They make it possible to form quotients, define simple Lie algebras, and build derived and central series.

The immediate test of this definition is whether it is exactly the condition needed for kernels and quotient constructions. If ideals are the correct analogue of normal subgroups, they should appear precisely as kernels of homomorphisms and precisely as the subspaces by which one can divide a Lie algebra.

[quotetheorem:3751]

[citeproof:3751]

This is the first indication that the usual algebraic tools still work in the Lie setting, provided multiplication is replaced by the bracket. The result also explains why ideals appear before more refined structure theory: every reduction to a quotient, every discussion of simplicity, and every passage to a smaller structural object depends on this compatibility condition rather than on internal closure alone. The point is not merely that ideals are useful examples of subspaces, but that they are the exact obstruction to transferring the bracket to a quotient. If a subspace is only closed under brackets among its own elements, representatives of the same coset may produce different bracket cosets; the extra ambient stability in the definition of an ideal is what removes this ambiguity.