This course develops the representation theory of Lie algebras as a systematic study of how abstract algebraic symmetries act on vector spaces. The central goal is to classify and compute representations, starting from the algebraic structure of universal enveloping algebras and the PBW theorem, then moving to weights, characters, and the special role of $\mathfrak{sl}_2$ as the basic model that guides the general theory. Along the way, the course emphasizes the mechanisms that make representations tractable: highest weight methods, Verma modules, central characters, and the structure of semisimple and reducible modules.

The chapters build in a deliberate progression. After establishing the enveloping algebra framework, the course uses $\mathfrak{sl}_2$ to motivate the language of weights and highest weight modules, then generalizes these ideas to the classification of irreducibles and the Weyl character formula. From there, it turns to tensor products, duals, invariants, and Schur functors as computational tools for building and decomposing representations, before studying branching rules and restriction to subalgebras. The final chapters place everything in the organizing framework of category $\mathcal O$ and synthesize the main results into a practical toolkit for both structural reasoning and explicit calculation.

# Introduction

This opening chapter fixes the viewpoint of the course and records the vocabulary that will be used throughout. The main object is a finite-dimensional complex semisimple [Lie algebra](/page/Lie%20Algebra) $\mathfrak{g}$, and the main question is how $\mathfrak{g}$ can act linearly on vector spaces. Earlier courses describe the internal structure of $\mathfrak{g}$ through Cartan subalgebras, roots, and the Weyl group; this course turns that structure into a classification and computation theory for representations.

The central theme is highest weight theory. Instead of trying to list all possible actions directly, we use the triangular decomposition of $\mathfrak{g}$ to build modules from a vector killed by all positive root spaces, then determine when those modules have finite-dimensional irreducible quotients. The final destination is the Weyl character formula, which packages the dimensions of all weight spaces of an irreducible finite-dimensional representation into a single computable expression.

## The Organising Question

A Lie algebra is designed to encode infinitesimal symmetry, so its representation theory asks how that infinitesimal symmetry appears inside linear algebra. For a fixed Lie algebra $\mathfrak{g}$, the raw data of a representation is a [vector space](/page/Vector%20Space) together with a compatible action of every element of $\mathfrak{g}$.

[definition: Representation Of A Lie Algebra] Let $\mathfrak{g}$ be a Lie algebra over $\mathbb C$. A representation of $\mathfrak{g}$ is a pair $(V,\rho)$ consisting of a complex vector space $V$ and a Lie algebra homomorphism \begin{align*} \rho : \mathfrak{g} \to \mathfrak{gl}(V). \end{align*} [/definition]

We often suppress $\rho$ and write $xv$ for $\rho(x)(v)$, where $x \in \mathfrak{g}$ and $v \in V$. With this notation, the defining compatibility condition says

\begin{align*} [x,y]v = x(yv)-y(xv) \end{align*}

for all $x,y \in \mathfrak{g}$ and $v \in V$.

The representation definition is efficient when we want to compare Lie algebras by homomorphisms into endomorphism algebras. For the rest of the course, however, we will constantly pass to subobjects, quotients, direct sums, tensor products, and generated cyclic objects. That makes it useful to restate the same data in module language, where those constructions are native.

[definition: Lie Algebra Module] Let $\mathfrak{g}$ be a Lie algebra over $\mathbb C$. A $\mathfrak{g}$-module is a complex vector space $V$ equipped with a bilinear map \begin{align*} \mathfrak{g}\times V \to V \end{align*} whose value at $(x,v)$ is denoted $xv$, such that \begin{align*} [x,y]v=x(yv)-y(xv) \end{align*} for all $x,y\in\mathfrak{g}$ and $v\in V$. [/definition]

Thus representations and modules are two languages for the same object. The representation language emphasises homomorphisms into endomorphism algebras, while the module language emphasises the internal structure of a vector space carrying an action.

[example: Natural Representation Of Sl Two] Let $V=\mathbb C^2$ with standard basis $v_1,v_2$. Define linear operators by \begin{align*} hv_1=v_1,\quad hv_2=-v_2,\quad ev_1=0,\quad ev_2=v_1,\quad fv_1=v_2,\quad fv_2=0. \end{align*} We check that these operators satisfy the $\mathfrak{sl}_2(\mathbb C)$ commutation relations. On $v_1$, \begin{align*} (he-eh)v_1=h(0)-e(v_1)=0=2ev_1. \end{align*} On $v_2$, \begin{align*} (he-eh)v_2=h(v_1)-e(-v_2)=v_1+v_1=2v_1=2ev_2. \end{align*} Thus $[h,e]=2e$. Similarly, \begin{align*} (hf-fh)v_1=h(v_2)-f(v_1)=-v_2-v_2=-2v_2=-2fv_1. \end{align*} and \begin{align*} (hf-fh)v_2=h(0)-f(-v_2)=0=-2fv_2. \end{align*} Thus $[h,f]=-2f$. Finally, \begin{align*} (ef-fe)v_1=e(v_2)-f(0)=v_1=hv_1. \end{align*} and \begin{align*} (ef-fe)v_2=e(0)-f(v_1)=-v_2=hv_2. \end{align*} Thus $[e,f]=h$, so the assignment $e\mapsto e$, $f\mapsto f$, $h\mapsto h$ gives the natural representation of $\mathfrak{sl}_2(\mathbb C)$ on $V$. The basis vectors are weight vectors for $h$: $v_1$ has weight $1$ and $v_2$ has weight $-1$. The operator $e$ raises $v_2$ to $v_1$ and kills $v_1$, while $f$ lowers $v_1$ to $v_2$ and kills $v_2$; hence $v_1$ is already a highest weight vector in this smallest nontrivial example. [/example]

The example shows why $\mathfrak{sl}_2(\mathbb C)$ remains the running test case. Many features of semisimple representation theory are visible there in compressed form, and the general theory can be read as a systematic extension of this pattern to all root systems.

## Why The Universal Enveloping Algebra Appears

A Lie algebra has no associative multiplication compatible with repeated actions on a module. Yet if $x,y\in\mathfrak{g}$ act on $V$, then the composite operator obtained by first applying $y$ and then applying $x$ is a genuine linear operator on $V$. Without an associative algebra containing these words, expressions such as $xyv$, $x^3yv$, or products of lowering operators have to be interpreted separately inside every representation. The universal enveloping algebra is introduced to keep track of all such iterated actions while imposing exactly the commutation relations required by the Lie bracket.

[definition: Universal Enveloping Algebra] Let $\mathfrak{g}$ be a Lie algebra over $\mathbb C$. The universal enveloping algebra $U(\mathfrak{g})$ is the associative unital algebra generated by $\mathfrak{g}$ subject to the relations \begin{align*} xy-yx=[x,y] \end{align*} for all $x,y\in\mathfrak{g}$. [/definition]

The first lecture makes this definition precise using the tensor algebra $T(\mathfrak{g})$ and a quotient by the two-sided ideal generated by $x\otimes y-y\otimes x-[x,y]$. A construction by generators and relations is only useful if it has the expected mapping property, so the next result identifies $U(\mathfrak{g})$ as the universal associative algebra through which all Lie algebra actions factor.