07.06.11 · representation-theory / lie-algebraic

Representations of $sl_{2} C$

shipped3 tiersLean: none

Anchor (Master): Cartan 1894 *Sur la structure des groupes de transformations finis et continus* (Thèse, Paris) — classification of simple Lie algebras with $\mathfrak{sl}_2$ as the base case; Weyl 1925-26 *Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen* (Math. Z. 23, 271–309; 24, 328–376, 377–395, 789–791) — unitarian trick and complete reducibility; Fulton-Harris *Representation Theory* §11 and §12 (ladder construction, Casimir, complete reducibility); Humphreys §7; Serre *Complex Semisimple Lie Algebras* §IV; Kac *Infinite Dimensional Lie Algebras* §1 (where the same ladder structure recurs at the level of affine Kac-Moody algebras)

Intuition [Beginner]

The Lie algebra $sl_{2} C$ is the smallest non-abelian Lie algebra that is rich enough to do interesting representation theory. It has three basis elements: a diagonal $H$ , a raising operator $E$ , and a lowering operator $F$ . The whole representation theory of every simple complex Lie algebra is built from copies of this one, so understanding what $sl_{2} C$ does on a vector space is the foundational example for the entire subject.

The picture is a ladder. The operator $H$ measures a number called the weight: each vector in the representation has a definite weight, and these weights are integers spaced two apart. The operator $E$ moves a vector up the ladder, adding two to its weight; the operator $F$ moves it down, subtracting two. A finite-dimensional representation is a finite ladder, with a top rung that $E$ kills (no higher rung exists) and a bottom rung that $F$ kills. The top rung determines everything: if its weight is $n$ , the ladder has $n + 1$ rungs and the weights are $n, n - 2, n - 4, \dots, - n$ .

Why does this matter? In quantum mechanics, the same ladder is the spin- $n /2$ particle: spin-zero is one rung, spin one-half is two rungs, spin one is three rungs. In algebra, every irreducible finite-dimensional representation of every simple complex Lie algebra (every type- $A$ , $B$ , $C$ , $D$ , $E$ , $F$ , $G$ classification piece) decomposes under suitable subalgebras into chains of these ladders. Get the $sl_{2}$ ladder right and the rest of representation theory unfolds from it.

Visual [Beginner]

A picture showing a vertical chain of weight spaces, each represented as a circle labelled with its $H$ -eigenvalue $n, n - 2, n - 4, \dots, - n$ from top to bottom. Arrows labelled $E$ point upward connecting each circle to the one above it (with $E$ killing the topmost circle), and arrows labelled $F$ point downward connecting each circle to the one below it (with $F$ killing the bottom circle). The arrows are decorated with the coefficient of the action — $F^{k} v_{n}$ on the right side of each level, and the upward action $E \cdot F^{k} v_{n} = k (n - k + 1) F^{k - 1} v_{n}$ recorded next to the upward arrow.

$A schematic ladder of weight spaces for the irreducible representation $V_n$ of $\mathfrak{sl}_2 \mathbb{C}$, with rungs labelled by the integer weights $n, n-2, \ldots, -n$, raising arrows $E$ moving up, and lowering arrows $F$ moving down.$

The picture captures the essential structure: a finite ladder of $n + 1$ rungs with $E$ raising and $F$ lowering. Every irreducible finite-dimensional representation of $sl_{2} C$ has exactly this shape, and the integer $n$ at the top determines the whole ladder.

Worked example [Beginner]

Build the irreducible representation $V_{3}$ of dimension $4$ — the spin- $3/2$ representation in physics language — and check the ladder structure on every basis vector.

Step 1. Start with the highest-weight vector $v_{3}$ . By construction $E$ kills it: $E v_{3} = 0$ . The eigenvalue of $H$ on $v_{3}$ is $3$ . So $H v_{3} = 3 v_{3}$ .

Step 2. Apply $F$ once to get $w_{1} = F v_{3}$ . The eigenvalue of $H$ on $w_{1}$ is $3 - 2 = 1$ , so $H w_{1} = w_{1}$ . The vector $w_{1}$ is the second rung of the ladder.

Step 3. Apply $F$ again: $w_{2} = F^{2} v_{3}$ . The $H$ -eigenvalue is $3 - 4 = - 1$ , so $H w_{2} = - w_{2}$ . Third rung.

Step 4. Apply $F$ once more: $w_{3} = F^{3} v_{3}$ . The $H$ -eigenvalue is $3 - 6 = - 3$ , so $H w_{3} = - 3 w_{3}$ . Fourth rung — the bottom.

Step 5. Check $F$ kills the bottom. $F w_{3} = F^{4} v_{3} = 0$ . The ladder stops at four rungs as predicted.

Step 6. Compute $E$ on each rung. The formula is $E \cdot F^{k} v_{3} = k (3 - k + 1) F^{k - 1} v_{3} = k (4 - k) F^{k - 1} v_{3}$ . Plugging in: $E w_{1} = 1 \cdot 3 \cdot v_{3} = 3 v_{3}$ ; $E w_{2} = 2 \cdot 2 \cdot w_{1} = 4 w_{1}$ ; $E w_{3} = 3 \cdot 1 \cdot w_{2} = 3 w_{2}$ .

What this tells us: the representation $V_{3}$ has dimension $4$ , basis ${v_{3}, w_{1}, w_{2}, w_{3}}$ , weights ${3, 1, - 1, - 3}$ , and the actions of $H$ , $E$ , $F$ are determined by simple integer formulas. Every irreducible representation of $sl_{2} C$ works the same way: choose the highest weight $n$ , build the chain $v_{n}, F v_{n}, F^{2} v_{n}, \dots, F^{n} v_{n}$ , and the entire structure is fixed.

Check your understanding [Beginner]

Formal definition [Intermediate+]

The Lie algebra $sl_{2} C$ is the three-dimensional complex Lie algebra of traceless $2 \times 2$ complex matrices under the commutator bracket. The standard basis is $$ H = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}, \qquad E = \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}, \qquad F = \begin{pmatrix} 0 & 0 \ 1 & 0 \end{pmatrix}, $$ with bracket relations $$ [H, E] = 2E, \qquad [H, F] = -2F, \qquad [E, F] = H. $$ A direct check confirms each: $H E - E H$ has off-diagonal entry $1 - (- 1) = 2$ in the $(1, 2)$ slot and zeroes elsewhere; $H F - F H$ has $(2, 1)$ entry $- 1 - 1 = - 2$ ; $E F - F E = diag (1, - 1) = H$ .

A representation of $sl_{2} C$ on a complex vector space $V$ is a Lie algebra homomorphism $ρ : sl_{2} C \to gl (V)$ , equivalently linear operators $ρ (H), ρ (E), ρ (F)$ on $V$ satisfying the three bracket relations. The representation is finite-dimensional when $V$ is finite-dimensional, irreducible when no proper non-zero subspace is invariant under $ρ (H)$ , $ρ (E)$ , $ρ (F)$ .

A weight vector of weight $λ \in C$ is a non-zero $v \in V$ with $ρ (H) v = λ v$ . The weight space $V_{λ} \subseteq V$ is the $λ$ -eigenspace of $ρ (H)$ . The bracket relations force the ladder action: if $v \in V_{λ}$ then $ρ (E) v \in V_{λ + 2}$ and $ρ (F) v \in V_{λ - 2}$ , since $$ \rho(H) \rho(E) v = \rho(E) \rho(H) v + [\rho(H), \rho(E)] v = \rho(E) (\lambda v) + 2 \rho(E) v = (\lambda + 2) \rho(E) v, $$ and similarly $ρ (F) v \in V_{λ - 2}$ .

A highest-weight vector of weight $λ$ is a weight vector $v_{λ} \neq = 0$ with $ρ (E) v_{λ} = 0$ . A highest-weight representation is one generated as a $sl_{2}$ -module by a highest-weight vector.

The Casimir element is $$ C = \rho(E) \rho(F) + \rho(F) \rho(E) + \tfrac{1}{2} \rho(H)^2 \in \mathrm{End}(V). $$ A short calculation using the bracket relations shows $C$ commutes with $ρ (H)$ , $ρ (E)$ , $ρ (F)$ , hence is a $sl_{2}$ -module endomorphism of $V$ . For irreducible $V$ , Schur's lemma forces $C$ to act as a scalar.

In typographic shorthand we drop $ρ$ and write $H v$ , $E v$ , $F v$ for the action on a vector, the convention used throughout. We write $V_{n}$ for the irreducible representation with highest weight $n$ , where $n$ is understood to be a non-negative integer.

Counterexamples to common slips

The bracket relation $[E, F] = H$ has a sign convention: with the basis $H, E, F$ chosen above, the relation is $E F - F E = H$ , not $F E - E F$ . Some texts use the opposite sign convention (taking $E$ and $F$ swapped); this flips the sign of $H$ but leaves the representation theory identical up to renaming.
The highest weight $n$ must be a non-negative integer for $V_{n}$ to be finite-dimensional. The construction $Verma (λ)$ makes sense for any $λ \in C$ but is infinite-dimensional unless $λ \in Z_{\geq 0}$ , in which case the unique irreducible quotient is $V_{λ} = V_{n}$ .
"Highest weight" refers to the top of the $H$ -eigenvalue ladder, not the largest eigenvalue in modulus. The lowest-weight vector $F^{n} v_{n}$ has $H$ -eigenvalue $- n$ , the smallest weight; both top and bottom are determined by $n$ .

Key theorem with proof [Intermediate+]

Theorem (Classification of finite-dimensional irreducible representations of $sl_{2} C$ ). For every non-negative integer $n$ , there exists a finite-dimensional irreducible representation $V_{n}$ of $sl_{2} C$ of dimension $n + 1$ , unique up to isomorphism, characterised by the following data. There is a basis $v_{0}^{(n)} = v_{n}, v_{1}^{(n)} = F v_{n}, v_{2}^{(n)} = F^{2} v_{n}, \dots, v_{n}^{(n)} = F^{n} v_{n}$ of $V_{n}$ on which the operators act by $$ H \cdot v_k^{(n)} = (n - 2k) v_k^{(n)}, \qquad F \cdot v_k^{(n)} = v_{k+1}^{(n)}, \qquad E \cdot v_k^{(n)} = k(n - k + 1) v_{k-1}^{(n)}, $$ with the conventions $v_{- 1}^{(n)} = 0$ and $v_{n + 1}^{(n)} = 0$ . Conversely, every finite-dimensional irreducible representation of $sl_{2} C$ is isomorphic to some $V_{n}$ .

Proof. The argument has three parts: existence, uniqueness of structure given a highest-weight vector, and the integrality $n \in Z_{\geq 0}$ .

Part 1: existence. Define $V_{n}$ as the vector space with basis symbols $v_{0}, v_{1}, \dots, v_{n}$ and declare $H$ , $E$ , $F$ to act by the stated formulas. The relations of $sl_{2} C$ must be checked on each basis vector.

For $[H, E] = 2 E$ , compute on $v_{k}$ : $$ (HE - EH) v_k = H \cdot k(n - k + 1) v_{k-1} - E \cdot (n - 2k) v_k = k(n - k + 1)(n - 2k + 2) v_{k-1} - (n - 2k) k(n - k + 1) v_{k-1}. $$ The right-hand side is $k (n - k + 1) [(n - 2 k + 2) - (n - 2 k)] v_{k - 1} = k (n - k + 1) \cdot 2 \cdot v_{k - 1} = 2 E v_{k}$ . Similarly $[H, F] v_{k} = (H F - F H) v_{k} = (n - 2 (k + 1)) v_{k + 1} - (n - 2 k) v_{k + 1} = - 2 v_{k + 1} = - 2 F v_{k}$ .

For $[E, F] = H$ : $$ (EF - FE) v_k = E v_{k+1} - F \cdot k(n - k + 1) v_{k-1} = (k + 1)(n - k) v_k - k(n - k + 1) v_k. $$ Expanding: $(k + 1) (n - k) - k (n - k + 1) = (k n - k^{2} + n - k) - (k n - k^{2} + k) = n - 2 k$ , which is the $H$ -eigenvalue on $v_{k}$ . So $[E, F] v_{k} = H v_{k}$ on every basis vector. The three relations hold, and $V_{n}$ is a valid representation. Irreducibility: any invariant subspace contains some $v_{k}$ (by triangularising under $H$ ), and iterating $E$ produces $v_{0}$ , iterating $F$ recovers all the other $v_{j}$ , so any non-zero invariant subspace is all of $V_{n}$ .

Part 2: structure from a highest-weight vector. Let $W$ be any finite-dimensional irreducible representation. The operator $H$ acts on $W$ , and since $W$ is finite-dimensional over the algebraically closed field $C$ , $H$ has an eigenvector. Choose any eigenvector $w$ of $H$ , with eigenvalue $μ$ . The chain $w, E w, E^{2} w, \dots$ has $H$ -eigenvalues $μ, μ + 2, μ + 4, \dots$ , all distinct, so the chain is linearly independent until it terminates. Since $W$ is finite-dimensional, the chain must terminate: there exists $j \geq 0$ with $E^{j} w \neq = 0$ and $E^{j + 1} w = 0$ . Set $v_{0} = E^{j} w$ and $λ = μ + 2 j$ . Then $v_{0}$ is a highest-weight vector of weight $λ$ .

Now define $v_{k} = F^{k} v_{0}$ for $k \geq 0$ . The $H$ -eigenvalue of $v_{k}$ is $λ - 2 k$ . We claim $$ E v_k = k(\lambda - k + 1) v_{k-1} \qquad \text{for } k \geq 1. $$ The proof is by induction on $k$ . Base case $k = 1$ : $E v_{1} = E F v_{0} = (F E + H) v_{0} = 0 + λ v_{0} = λ v_{0} = 1 \cdot (λ - 0) v_{0}$ . Inductive step: assuming the formula for $k$ , compute $E v_{k + 1} = E F v_{k} = F E v_{k} + H v_{k} = F \cdot k (λ - k + 1) v_{k - 1} + (λ - 2 k) v_{k} = k (λ - k + 1) v_{k} + (λ - 2 k) v_{k} = [k (λ - k + 1) + (λ - 2 k)] v_{k}$ . Simplify the bracket: $k λ - k^{2} + k + λ - 2 k = (k + 1) λ - k^{2} - k = (k + 1) (λ - k)$ . So $E v_{k + 1} = (k + 1) (λ - k) v_{k}$ , which is the formula with $k$ replaced by $k + 1$ . The induction is complete.

Part 3: integrality. The chain $v_{0}, v_{1}, v_{2}, \dots$ has linearly independent terms (different $H$ -eigenvalues) until it terminates. By finite-dimensionality, there is a smallest $N \geq 0$ with $v_{N + 1} = F^{N + 1} v_{0} = 0$ but $v_{N} \neq = 0$ . Apply $E$ : by the formula, $0 = E v_{N + 1} = (N + 1) (λ - N) v_{N}$ . Since $v_{N} \neq = 0$ and $N + 1 \neq = 0$ , we conclude $λ - N = 0$ , that is, $λ = N$ . So $λ$ is a non-negative integer, and the chain has $λ + 1 = N + 1$ non-zero terms. The span of $v_{0}, v_{1}, \dots, v_{N}$ is invariant under $H$ , $E$ , $F$ (by the formulas), non-zero, so equal to $W$ by irreducibility. Setting $n = λ = N$ , the operators $H$ , $E$ , $F$ act on this basis by exactly the formulas of the theorem. So $W ≅ V_{n}$ and the classification is complete. $□$

Bridge. This classification builds toward the full Cartan-Killing-Weyl theory of semisimple Lie algebras, where the same ladder structure appears again in 07.06.03 (root system) decorating each root by an $sl_{2}$ -triple $(H_{α}, E_{α}, F_{α})$ . The foundational reason the highest-weight theorem works on $sl_{2} C$ is exactly the relation $E v_{1} = λ v_{0}$ together with the inductive formula $E v_{k + 1} = (k + 1) (λ - k) v_{k}$ , which forces the chain to terminate at $λ = N$ and identifies the irreducible with the integer $N$ . The central insight is that an irreducible finite-dimensional representation of $sl_{2} C$ is uniquely determined by its highest weight $n \in Z_{\geq 0}$ , and the operators $H$ , $E$ , $F$ act by explicit integer-coefficient formulas on the ladder basis $v_{n}, F v_{n}, \dots, F^{n} v_{n}$ . This is exactly the structural content that generalises to the Cartan-Weyl classification: every irreducible representation of a semisimple complex Lie algebra is generated by a highest-weight vector and parametrised by an integral dominant weight, and the ladder calculation we ran here is the rank-one engine. The bridge is the recognition that the $sl_{2}$ -ladder is the local model for every weight chain in every higher-rank Lie algebra. Putting these together, the Casimir element $C = E F + F E + H^{2} /2$ identifies $V_{n}$ with the scalar $n (n + 2) /2$ — appears again in 07.06.07 (Weyl character formula) as the rank-one input to the universal character identity, and generalises in higher rank to the full Harish-Chandra centre of the universal enveloping algebra.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

Realise $V_{n}$ explicitly as the symmetric power $Sym^{n} (C^{2})$ , with $sl_{2} C$ acting on $C^{2}$ in the defining representation and extended to the symmetric power by the Leibniz rule. Identify the highest-weight vector and the action of $H$ , $E$ , $F$ on the monomial basis.

Hint

Let $x, y$ be a basis of $C^{2}$ with $H$ -weights $+ 1, - 1$ . Then $Sym^{n} (C^{2})$ has basis ${x^{n}, x^{n - 1} y, \dots, y^{n}}$ , and $E, F$ act as derivations: $E = x \partial_{y}$ , $F = y \partial_{x}$ , $H = x \partial_{x} - y \partial_{y}$ .

Answer

Take $C^{2}$ with basis $x, y$ on which $H$ acts diagonally with eigenvalues $H x = x$ , $H y = - y$ , and $E$ , $F$ act by $E x = 0$ , $E y = x$ , $F x = y$ , $F y = 0$ . Extend to $Sym^{n} (C^{2}) = C [x, y]_{n}$ (homogeneous polynomials of degree $n$ ) by the derivation rule. Then $H = x \partial_{x} - y \partial_{y}$ , $E = x \partial_{y}$ , $F = y \partial_{x}$ . The monomial $x^{n}$ has $H$ -eigenvalue $n$ (highest weight) and is killed by $E$ (no $y$ to derive). Apply $F = y \partial_{x}$ : $F (x^{n}) = n x^{n - 1} y$ , then $F^{2} (x^{n}) = n (n - 1) x^{n - 2} y^{2}$ , in general $F^{k} (x^{n}) = n (n - 1) \dots (n - k + 1) x^{n - k} y^{k} = \frac{n !}{( n - k )!} x^{n - k} y^{k}$ . Comparing with the canonical ladder basis $v_{k} = F^{k} v_{n} /1$ (here we choose $v_{n} = x^{n}$ ), the identification is $v_{k} = \frac{n !}{( n - k )!} x^{n - k} y^{k}$ . The $H$ -eigenvalue of $x^{n - k} y^{k}$ is $(n - k) - k = n - 2 k$ , matching the formula. Check $E$ : $E (x^{n - k} y^{k}) = x \partial_{y} (x^{n - k} y^{k}) = k x^{n - k + 1} y^{k - 1}$ , which under the identification $v_{k - 1} = \frac{n !}{( n - k + 1 )!} x^{n - k + 1} y^{k - 1}$ gives $E v_{k} = k \cdot \frac{( n - k + 1 )!}{n !} \cdot x^{n - k + 1} y^{k - 1} \cdot \frac{n !}{( n - k )!} = k (n - k + 1) v_{k - 1}$ , recovering the ladder formula. Rubric: full credit for identifying $Sym^{n} (C^{2})$ with $V_{n}$ via the polynomial differential operators and checking the ladder action.

Exercise 7 (hard, short-answer).

State and prove the general Clebsch-Gordan formula $V_{m} \otimes V_{n} = ⨁_{k = 0}^{m i n (m, n)} V_{m + n - 2 k}$ for $sl_{2} C$ .

Hint

Count weight multiplicities in the tensor product, then peel off the highest-weight irreducible at each stage. The weight multiplicity of $j$ in $V_{m} \otimes V_{n}$ counts pairs $(a, b)$ with $a + b = j$ , $∣ a ∣ \leq m$ , $∣ b ∣ \leq n$ , and $a \equiv m mod 2$ , $b \equiv n mod 2$ .

Answer

Without loss of generality assume $m \leq n$ . The weight multiplicities of $V_{m} \otimes V_{n}$ are computed by convolving the multiplicity functions of the two factors: weight $j$ in $V_{m} \otimes V_{n}$ has multiplicity equal to the number of pairs $(a, b)$ with $a$ a weight of $V_{m}$ , $b$ a weight of $V_{n}$ , $a + b = j$ . The weight set of $V_{m}$ is ${- m, - m + 2, \dots, m - 2, m}$ ( $m + 1$ values, all of parity $m mod 2$ ). Similarly for $V_{n}$ .

The convolution multiplicity at weight $j$ is the number of ways to write $j = a + b$ with $a, b$ in those sets. A short calculation gives multiplicity $min (m, n) + 1 - ⌈(∣ j ∣ - ∣ m - n ∣) /2 ⌉$ when $∣ j ∣ \leq m + n$ and $j \equiv m + n (mod 2)$ , else zero.

Now peel off the highest-weight irreducible: the highest weight present is $m + n$ with multiplicity $1$ , contributing a $V_{m + n}$ summand with weights ${m + n, m + n - 2, \dots, - (m + n)}$ each of multiplicity $1$ . Subtract: the residual multiplicities at weights $j \in {m + n - 2, m + n - 4, \dots}$ are reduced by $1$ . The new highest weight is $m + n - 2$ (with reduced multiplicity at least $1$ when $min (m, n) \geq 1$ ), giving a $V_{m + n - 2}$ summand. Iterate.

Each step reduces the highest weight by $2$ and the total dimension by the dimension of the peeled summand. The process terminates after $min (m, n) + 1$ steps, yielding $V_{m} \otimes V_{n} = V_{m + n} \oplus V_{m + n - 2} \oplus \dots \oplus V_{∣ m - n ∣}$ . Equivalently, $⨁_{k = 0}^{m i n (m, n)} V_{m + n - 2 k}$ . Dimension check: $\sum_{k = 0}^{m i n (m, n)} (m + n - 2 k + 1) = (min (m, n) + 1) (m + n - min (m, n) + 1) = (m + 1) (n + 1)$ , matching $dim V_{m} \cdot dim V_{n}$ . Rubric: full credit for the formula plus the weight-multiplicity-convolution argument.

Exercise 8 (hard, short-answer).

Prove Weyl's complete-reducibility theorem for $sl_{2} C$ : every finite-dimensional representation of $sl_{2} C$ is a direct sum of irreducibles $V_{n}$ .

Hint

Reduce to showing: every short exact sequence $0 \to W \to V \to V / W \to 0$ of $sl_{2}$ -modules splits. Use the Casimir element to construct a splitting, separating summands by Casimir eigenvalues.

Answer

It suffices to show every short exact sequence $0 \to W \to V \to V / W \to 0$ of finite-dimensional $sl_{2}$ -modules splits; iterating builds a direct-sum decomposition into irreducibles. By a standard reduction (replace $V$ by a suitable extension if necessary), we may assume $W$ and $V / W$ are irreducibles $V_{m}$ and $V_{n}$ .

Case 1: $m \neq = n$ . The Casimir $C$ acts as $m (m + 2) /2$ on $W$ and as $n (n + 2) /2$ on $V / W$ , and these scalars are distinct (the function $x \mapsto x (x + 2) /2$ is strictly increasing on $[0, \infty)$ ). So $C$ on $V$ has the polynomial $(C - m (m + 2) /2) (C - n (n + 2) /2)$ as a vanishing element, with kernel decomposition $V = ker (C - m (m + 2) /2) \oplus ker (C - n (n + 2) /2) = W \oplus V^{'}$ , where $V^{'} ≅ V / W$ provides the splitting.

Case 2: $m = n$ , so $W ≅ V_{n}$ and $V / W ≅ V_{n}$ . The Casimir scalar agrees on both, and a direct argument is needed. One classical route (Weyl): construct a $sl_{2}$ -invariant Hermitian inner product on $V$ using the unitarian trick — the compact real form $su (2)$ exponentiates to the compact group $SU (2)$ , every finite-dimensional representation of $SU (2)$ admits an invariant Hermitian inner product by averaging, and this restricts to an $sl_{2} C$ -invariant inner product on $V$ . The orthogonal complement $W^{⊥}$ is a $sl_{2}$ -invariant complement of $W$ , giving the splitting $V = W \oplus W^{⊥}$ . Alternatively: a cohomological argument shows $Ext_{sl_{2} C}^{1} (V_{n}, V_{n}) = 0$ for finite-dimensional modules. Both routes give the splitting. Rubric: full credit for the Casimir argument in the unequal case plus the unitarian-trick (or cohomological) argument in the equal case.

Lean formalization [Intermediate+]

Mathlib has the abstract Lie-algebra infrastructure in Mathlib.Algebra.Lie.Basic, the matrix-bracket bridge in Mathlib.Algebra.Lie.OfAssociative, and weight-module scaffolding in Mathlib.Algebra.Lie.Weights, but no named development of $sl_{2} C$ irreducible representations. The intended formalisation reads schematically:

import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.LinearAlgebra.Matrix.Trace

namespace Codex.Sl2

/-- The Lie algebra sl₂ ℂ as traceless 2×2 complex matrices. -/
abbrev sl2 : Type := { M : Matrix (Fin 2) (Fin 2) ℂ // M.trace = 0 }

/-- The standard generator H = diag(1, -1). -/
noncomputable def H : sl2 := sorry

/-- The raising operator E. -/
noncomputable def E : sl2 := sorry

/-- The lowering operator F. -/
noncomputable def F : sl2 := sorry

/-- Bracket relation [H, E] = 2 E. -/
theorem bracket_H_E : ⁅H, E⁆ = (2 : ℂ) • E := sorry

/-- Bracket relation [H, F] = -2 F. -/
theorem bracket_H_F : ⁅H, F⁆ = (-2 : ℂ) • F := sorry

/-- Bracket relation [E, F] = H. -/
theorem bracket_E_F : ⁅E, F⁆ = H := sorry

/-- The irreducible representation V_n of dimension n+1. -/
noncomputable def V (n : ℕ) : Type := Fin (n + 1) → ℂ

/-- Highest-weight theorem: every finite-dimensional irreducible
    representation of sl₂ ℂ is isomorphic to V_n for some n. -/
theorem highestWeight_classification
    {W : Type*} [AddCommGroup W] [Module ℂ W] [Finite W]
    [LieModule ℂ sl2 W] (irred : IsIrreducible ℂ sl2 W) :
    ∃ n : ℕ, Nonempty (W ≃ₗ⁅sl2⁆ V n) :=
  sorry  -- existence of highest-weight vector + ladder construction

/-- Casimir scalar on V_n. -/
theorem casimir_eigenvalue (n : ℕ) (v : V n) :
    (E * F + F * E + (1/2 : ℂ) • H * H) • v = (n * (n + 2) / 2 : ℂ) • v :=
  sorry

end Codex.Sl2

The proof gap is finite-dimensional linear algebra: the ladder formula $E \cdot F^{k} v_{n} = k (n - k + 1) F^{k - 1} v_{n}$ , the irreducibility of the span, the integrality argument forcing $n \in Z_{\geq 0}$ , and the Casimir-scalar calculation are each straightforward computations that Mathlib's matrix-and-module infrastructure supports, but none have been packaged as named theorems. Complete reducibility for $sl_{2} C$ requires either the unitarian-trick route through SU(2) (Mathlib has Matrix.SpecialUnitaryGroup; the averaging-over-the-compact-group construction is not packaged) or the cohomological route through Ext-groups for Lie-algebra modules (also not packaged). Either path is a formalisation target, and the scaffolding to attempt them exists.

Advanced results [Master]

Theorem (Symmetric-power realisation). For every non-negative integer $n$ , the symmetric power $Sym^{n} (C^{2})$ of the defining representation of $sl_{2} C$ on $C^{2}$ is isomorphic to $V_{n}$ . Choosing a basis $x, y$ of $C^{2}$ with $H$ -eigenvalues $+ 1, - 1$ , the polynomial ring $C [x, y]_{n}$ (homogeneous degree $n$ ) carries the $sl_{2} C$ action $H = x \partial_{x} - y \partial_{y}$ , $E = x \partial_{y}$ , $F = y \partial_{x}$ , with highest-weight vector $x^{n}$ and ladder basis ${x^{n}, x^{n - 1} y, x^{n - 2} y^{2}, \dots, y^{n}}$ (up to rescaling).

This is the most natural geometric realisation of $V_{n}$ : every irreducible representation of $sl_{2} C$ appears as a space of polynomials, and the operators are first-order differential operators. The same construction generalises to $sl_{n} C$ via Schur-Weyl duality.

Theorem (Decomposition of tensor products; Clebsch-Gordan rule). For all non-negative integers $m, n$ , $$ V_m \otimes V_n \cong \bigoplus_{k = 0}^{\min(m, n)} V_{m + n - 2k} = V_{m + n} \oplus V_{m + n - 2} \oplus \cdots \oplus V_{|m - n|}. $$ The decomposition is unique up to isomorphism and is computed by weight-multiplicity convolution.

The Clebsch-Gordan rule is the rank-one ancestor of the Littlewood-Richardson rule for $sl_{n} C$ . Its formal statement appeared in Clebsch and Gordan's 1860s work on invariants of binary forms, long before Lie theory; the modern interpretation is that binary forms of degree $n$ are exactly the $V_{n}$ representations, and their product decompositions are tensor-product decompositions.

Theorem (Symmetric and exterior powers). For non-negative integers $n, k$ , $$ \mathrm{Sym}^k(V_n) = \bigoplus_{j} V_{(k - 2j) n - 2 \cdot \text{(combinatorial term)}} $$ with explicit multiplicities computable via Hilbert-Poincaré series, and similarly for exterior powers $Λ^{k} (V_{n})$ . The two are related by the involution $\mathrm{Sym}^k(V_n) \leftrightarrow \Lambda^k(V_n^)$ on parity.*

The explicit decompositions are not as compact as Clebsch-Gordan but follow from the same weight-counting machinery. They are the building blocks for computing every plethysm operation on $sl_{2} C$ representations.

Theorem (Weyl's complete reducibility, $sl_{2}$ case). Every finite-dimensional representation of $sl_{2} C$ is isomorphic to a direct sum of irreducibles $V_{n}$ , with the multiplicities determined by the Casimir-eigenvalue decomposition together with the weight-multiplicity structure.

The classical proof routes are the unitarian trick (the compact real form $su (2)$ exponentiates to $SU (2)$ , every finite-dimensional representation of which admits an invariant Hermitian inner product by averaging over the compact group, giving orthogonal complements as $sl_{2}$ -invariant splittings), the Casimir splitting (the operator $C$ has distinct eigenvalues $n (n + 2) /2$ on distinct irreducibles, so its generalised eigenspaces decompose any module into Casimir-isotypic components, after which the equal-Casimir case is handled separately), and the cohomological splitting (vanishing of $Ext_{sl_{2}}^{1} (V_{m}, V_{n})$ for finite-dimensional modules by an explicit Casimir-based homotopy argument).

Theorem (Centre of $U (sl_{2} C)$ ). The centre of the universal enveloping algebra $U (sl_{2} C)$ is the polynomial algebra $C [C]$ generated by the Casimir element $C = E F + F E + H^{2} /2$ . The Harish-Chandra isomorphism identifies the centre with the Weyl-invariant polynomials on the dual of the Cartan subalgebra: $C [H]^{W} = C [H^{2}]$ , where the Weyl group $W = Z /2$ acts by $H \mapsto - H$ , and $C \leftrightarrow H^{2} /2 + H$ under the appropriate identification.

This is the rank-one instance of Harish-Chandra's theorem on the centre of the universal enveloping algebra of a semisimple Lie algebra. The Casimir's centrality is what powers complete reducibility, the Verma-module structure (see 07.06.06), and the constructive parts of the highest-weight theorem.

Theorem (Connection to $SU (2)$ representations). Every finite-dimensional complex-linear representation of $sl_{2} C$ comes from a unique representation of $SL_{2} C$ (the complex Lie group), and restricting to the compact real form $SU (2) \subseteq SL_{2} C$ gives a one-to-one correspondence between finite-dimensional complex representations of $sl_{2} C$ and finite-dimensional complex representations of $SU (2)$ . Under this correspondence, $V_{n}$ is the spin- $n /2$ representation in physics terminology, and the integer $n$ is twice the spin.

This is the unitarian trick at the level of finite-dimensional representations, originally due to Weyl 1925. The compact group $SU (2)$ is connected, simply connected, and compact, so its representations are completely understood through compact-group representation theory (Peter-Weyl). Transporting back to $sl_{2} C$ via the Lie algebra of the complexification recovers the classification above. The integer $n$ corresponds to spin $j = n /2$ , with $V_{n}$ having dimension $2 j + 1$ .

Theorem (Crystal limit and Lusztig-Kashiwara). In the quantum-group setting $U_{q} (sl_{2})$ , the representations $V_{n}^{q}$ deform the $V_{n}$ with the same dimensions. The crystal basis of $V_{n}^{q}$ (Kashiwara 1991) is a combinatorial skeleton: the basis vectors $b_{0}, b_{1}, \dots, b_{n}$ form a single chain with crystal operators $\tilde{E} (b_{k}) = b_{k - 1}$ and $\tilde{F} (b_{k}) = b_{k + 1}$ (and zero off the chain). At $q = 1$ the crystal basis recovers the classical ladder basis up to normalisation.

The crystal basis is the combinatorial shadow of the representation, extracted by passing to a degeneration limit of the quantum group. For $sl_{2}$ the crystals are particularly clean: a single chain with two arrows per node. This shadow encodes the entire representation theory and generalises to higher-rank Kac-Moody algebras.

Synthesis. The classification of irreducible $sl_{2} C$ -representations $V_{n}$ (one per non-negative integer, with explicit ladder action) is the foundational reason that the Cartan-Killing-Weyl theory of complex semisimple Lie algebras works at all. The central insight is that every irreducible representation is generated by a highest-weight vector $v_{n}$ annihilated by $E$ with $H$ -eigenvalue $n \in Z_{\geq 0}$ , and the chain $v_{n}, F v_{n}, \dots, F^{n} v_{n}$ exhausts the representation with weights $n, n - 2, \dots, - n$ . This is exactly the local model for every weight chain in every higher-rank Lie algebra: each root $α$ of a semisimple Lie algebra $g$ gives an $sl_{2}$ -triple $(H_{α}, E_{α}, F_{α})$ , and the representation theory of $g$ on any module restricts along each such triple to a direct sum of $V_{n}$ 's. The integrality of weights in higher rank, the Weyl-group symmetry of weight diagrams, the dominance condition, and the Weyl dimension formula all generalise the $sl_{2}$ ladder calculation we ran. Putting these together, every theorem in the structure theory of finite-dimensional representations of semisimple Lie algebras is, at heart, an $sl_{2}$ ladder computation indexed by a root system. The bridge between the algebraic and the analytic side of the story is the unitarian trick: the same $V_{n}$ is realised as the spin- $n /2$ representation of $SU (2)$ , where averaging over the compact group yields invariant Hermitian inner products and the proof of complete reducibility runs through compact-group representation theory rather than through Lie-algebra cohomology.

The duality between the ladder picture and the geometric picture is precise. The symmetric-power realisation $V_{n} = Sym^{n} (C^{2})$ identifies $V_{n}$ with the space of homogeneous polynomials of degree $n$ on which $E = x \partial_{y}$ , $F = y \partial_{x}$ , $H = x \partial_{x} - y \partial_{y}$ act as first-order differential operators. The same representation is dual to the Verma module $M (n)$ (see 07.06.06) modulo its maximal proper submodule, giving the Borel-Weil realisation 07.06.09 of $V_{n}$ as sections of a line bundle on $C P^{1} = SU (2) / T$ in degree $n$ . The Casimir $C = E F + F E + H^{2} /2$ acts by $n (n + 2) /2$ and identifies the irreducible by its eigenvalue, recurring in higher rank as the quadratic Casimir indexing irreducible representations by their infinitesimal character. The Clebsch-Gordan rule $V_{m} \otimes V_{n} = ⨁ V_{m + n - 2 k}$ is the rank-one Littlewood-Richardson rule; the symmetric-power and exterior-power decompositions are the rank-one plethysm; the centre $C [C]$ of $U (sl_{2} C)$ is the rank-one Harish-Chandra centre. Quantum deformation produces $U_{q} (sl_{2})$ and crystal bases ${b_{0}, \dots, b_{n}}$ that encode the representation theory combinatorially; affine deformation produces $sl_{2}$ with infinite-dimensional integrable representations whose level-and-weight structure recapitulates the $sl_{2}$ ladder at every node of an extended Dynkin diagram. The bridge is that every refinement of representation theory — quantum, affine, supersymmetric, $p$ -adic — preserves the rank-one ladder as the local model.

Full proof set [Master]

Theorem (classification of irreducible $sl_{2} C$ -representations), proof. Given in full in the Intermediate-tier section: a highest-weight vector $v_{0}$ of weight $λ$ exists by finite-dimensionality and algebraic closure; the chain $F^{k} v_{0}$ has $H$ -eigenvalues $λ - 2 k$ and is independent until $F^{N + 1} v_{0} = 0$ ; the formula $E F^{k} v_{0} = k (λ - k + 1) F^{k - 1} v_{0}$ holds by induction; applying $E$ to $F^{N + 1} v_{0} = 0$ forces $(N + 1) (λ - N) = 0$ , hence $λ = N \in Z_{\geq 0}$ . Irreducibility of $V_{n}$ follows because any non-zero invariant subspace contains some weight vector $F^{k} v_{n}$ , and applying $E$ and $F$ recovers the rest. Uniqueness up to isomorphism: any two irreducibles with the same highest weight $n$ have isomorphic ladder bases, and the formulas determine the actions of $H, E, F$ on those bases. $□$

Proposition (Casimir centrality). The element $C = E F + F E + H^{2} /2 \in U (sl_{2} C)$ lies in the centre of the universal enveloping algebra. On the finite-dimensional irreducible $V_{n}$ , $C$ acts as the scalar $n (n + 2) /2$ .

Proof. Compute the three commutators in $U (sl_{2} C)$ , using the defining relations $[H, E] = 2 E$ , $[H, F] = - 2 F$ , $[E, F] = H$ . For $[H, C]$ : $[H, E F] = [H, E] F + E [H, F] = 2 E F + E (- 2 F) = 2 E F - 2 E F = 0$ . Similarly $[H, F E] = - 2 F E + 2 F E = 0$ , and $[H, H^{2}] = 0$ . So $[H, C] = 0$ .

For $[E, C]$ : $[E, E F] = E [E, F] = E H$ . And $[E, F E] = [E, F] E = H E$ . For $[E, H^{2} /2] = \frac{1}{2} ([E, H] H + H [E, H]) = \frac{1}{2} (- 2 E \cdot H + H \cdot (- 2 E)) = - (E H + H E)$ . Sum: $[E, C] = E H + H E - (E H + H E) = 0$ .

For $[F, C]$ by symmetry: $[F, E F] = [F, E] F = - H F$ . And $[F, F E] = F [F, E] = - F H$ . For $[F, H^{2} /2] = \frac{1}{2} ([F, H] H + H [F, H]) = \frac{1}{2} (2 F H + H \cdot 2 F) = F H + H F$ . Sum: $[F, C] = - H F - F H + F H + H F = 0$ .

So $C$ commutes with each of $H, E, F$ , hence with all of $sl_{2} C$ , hence with all of $U (sl_{2} C)$ since $U$ is generated by $sl_{2} C$ .

To compute the Casimir scalar on $V_{n}$ , evaluate $C$ on the highest-weight vector $v_{n}$ . Since $E v_{n} = 0$ , we have $F E v_{n} = 0$ and $E F v_{n} = (F E + H) v_{n} = H v_{n} = n v_{n}$ . So $C v_{n} = n v_{n} + 0 + (n^{2} /2) v_{n} = (n + n^{2} /2) v_{n} = n (n + 2) /2 \cdot v_{n}$ . By Schur's lemma applied to the irreducible $V_{n}$ , $C$ acts as this same scalar on all of $V_{n}$ . $□$

Proposition (symmetric-power realisation). The action $H = x \partial_{x} - y \partial_{y}$ , $E = x \partial_{y}$ , $F = y \partial_{x}$ on $Sym^{n} (C^{2}) = C [x, y]_{n}$ makes $Sym^{n} (C^{2})$ into an irreducible $sl_{2} C$ -representation isomorphic to $V_{n}$ .

Proof. First verify the bracket relations. $[H, E] = H E - E H = (x \partial_{x} - y \partial_{y}) (x \partial_{y}) - (x \partial_{y}) (x \partial_{x} - y \partial_{y})$ . Compute term-by-term using $[\partial_{x}, x] = 1$ , $[\partial_{y}, y] = 1$ : $(x \partial_{x}) (x \partial_{y}) = x (\partial_{x} x) \partial_{y} = x (1 + x \partial_{x}) \partial_{y} = x \partial_{y} + x^{2} \partial_{x} \partial_{y}$ . Similarly $(x \partial_{y}) (x \partial_{x}) = x^{2} \partial_{y} \partial_{x} = x^{2} \partial_{x} \partial_{y}$ . And $(y \partial_{y}) (x \partial_{y}) = x y \partial_{y} \partial_{y}$ , $(x \partial_{y}) (y \partial_{y}) = x (\partial_{y} y) \partial_{y} = x (1 + y \partial_{y}) \partial_{y} = x \partial_{y} + x y \partial_{y}^{2}$ . Combining: $H E = (x \partial_{x}) (x \partial_{y}) - (y \partial_{y}) (x \partial_{y}) = x \partial_{y} + x^{2} \partial_{x} \partial_{y} - x y \partial_{y}^{2}$ . And $E H = (x \partial_{y}) (x \partial_{x}) - (x \partial_{y}) (y \partial_{y}) = x^{2} \partial_{x} \partial_{y} - x \partial_{y} - x y \partial_{y}^{2}$ . So $[H, E] = H E - E H = x \partial_{y} - (- x \partial_{y}) = 2 x \partial_{y} = 2 E$ . Similarly $[H, F] = - 2 F$ and $[E, F] = H$ . The relations hold.

The highest-weight vector is $x^{n}$ : $E (x^{n}) = x \partial_{y} (x^{n}) = 0$ since $x^{n}$ has no $y$ . And $H (x^{n}) = x \partial_{x} (x^{n}) - y \partial_{y} (x^{n}) = n x^{n} - 0 = n x^{n}$ . So $x^{n}$ is a highest-weight vector of weight $n$ .

The ladder applied to $x^{n}$ : $F (x^{n}) = y \partial_{x} (x^{n}) = n y x^{n - 1}$ , then $F^{2} (x^{n}) = y \partial_{x} (n y x^{n - 1}) = n (n - 1) y^{2} x^{n - 2}$ , in general $F^{k} (x^{n}) = \frac{n !}{( n - k )!} y^{k} x^{n - k}$ , terminating at $F^{n + 1} (x^{n}) = 0$ . The chain spans $C [x, y]_{n}$ , which has dimension $n + 1$ , so $Sym^{n} (C^{2})$ is exactly the ladder representation $V_{n}$ . Irreducibility follows from the chain spanning. $□$

Proposition (Clebsch-Gordan formula). For non-negative integers $m, n$ , the tensor product decomposes as $V_{m} \otimes V_{n} = V_{m + n} \oplus V_{m + n - 2} \oplus \dots \oplus V_{∣ m - n ∣}$ .

Proof. Without loss of generality $m \leq n$ . Compute the weight-multiplicity generating polynomial. The character (formal sum of weights with multiplicities) of $V_{n}$ is $χ_{n} (t) = t^{n} + t^{n - 2} + \dots + t^{- n} = \sum_{k = 0}^{n} t^{n - 2 k}$ . The character of a tensor product is the product of characters: $$ \chi_m(t) \chi_n(t) = \left(\sum_{a = 0}^m t^{m - 2a}\right)\left(\sum_{b = 0}^n t^{n - 2b}\right). $$ We compare this to a candidate decomposition: $\sum_{k = 0}^{m} χ_{m + n - 2 k} (t)$ . Expand $χ_{m + n - 2 k} (t) = \sum_{j = 0}^{m + n - 2 k} t^{m + n - 2 k - 2 j}$ . The double sum is $$ \sum_{k = 0}^m \sum_{j = 0}^{m + n - 2k} t^{m + n - 2k - 2j}. $$ Substituting $i = k + j$ (so $i$ ranges over ${k, k + 1, \dots, m + n - k}$ ), this becomes $\sum_{k = 0}^{m} \sum_{i = k}^{m + n - k} t^{m + n - 2 i}$ . Reorder the sum to be indexed by $i$ first: each $t^{m + n - 2 i}$ appears once for every $k \in {0, 1, \dots, min (i, m + n - i)}$ with $k \leq m$ . For $i \in {0, 1, \dots, m + n}$ the count is $min (i, m + n - i, m) + 1$ .

A short check confirms this matches the product $χ_{m} (t) χ_{n} (t)$ : in the product, the coefficient of $t^{m + n - 2 i}$ is the number of pairs $(a, b)$ with $a + b = i$ , $0 \leq a \leq m$ , $0 \leq b \leq n$ , which is $min (i, m + n - i, m, n) + 1 = min (i, m + n - i, m) + 1$ since $m \leq n$ . So the characters agree, and by character-theoretic determination of finite-dimensional representations of $sl_{2} C$ (the irreducibles are pairwise non-isomorphic and characters of direct sums are sums of characters), $V_{m} \otimes V_{n} = ⨁_{k = 0}^{m} V_{m + n - 2 k}$ . $□$

Proposition (Weyl complete reducibility for $sl_{2} C$ ). Every finite-dimensional representation $V$ of $sl_{2} C$ is a direct sum of irreducibles $V_{n}$ .

Proof (unitarian trick). Consider the compact real form $su (2) \subseteq sl_{2} C$ , spanned by $i H$ , $E - F$ , $i (E + F)$ . The Lie group integrating $su (2)$ is $SU (2)$ , which is connected, simply connected, and compact. The complex-linear representation $V$ of $sl_{2} C$ restricts to a real Lie algebra representation of $su (2)$ , which exponentiates uniquely (using simply-connectedness) to a representation of $SU (2)$ on $V$ . By averaging an arbitrary Hermitian inner product on $V$ over $SU (2)$ using Haar measure (Haar measure exists by compactness and is unique up to normalisation), we obtain an $SU (2)$ -invariant positive-definite Hermitian inner product $⟨ \cdot, \cdot ⟩$ on $V$ , hence an $su (2)$ -invariant inner product, hence (by $C$ -linear extension of the relations) an $sl_{2} C$ -invariant inner product on $V$ .

Now suppose $W \subseteq V$ is an $sl_{2} C$ -invariant subspace. The orthogonal complement $W^{⊥}$ with respect to the invariant inner product is also $sl_{2} C$ -invariant, because if $w \in W^{⊥}$ and $X \in sl_{2} C$ then $⟨ X w, w^{'} ⟩ = - ⟨ w, X^{*} w^{'} ⟩ = 0$ for all $w^{'} \in W$ (using that $X^{*} = - X$ for $X \in su (2)$ and extending by $C$ -linearity). So $V = W \oplus W^{⊥}$ as $sl_{2} C$ -modules. Iterating, $V$ decomposes into a direct sum of irreducible $sl_{2} C$ -submodules, each isomorphic to some $V_{n}$ by the classification. $□$

Connections [Master]

Lie algebra representation 07.06.01. The general definition of a Lie algebra representation specialises here to the rank-one case $sl_{2} C$ , where the entire classification is explicit: one irreducible $V_{n}$ per non-negative integer, dimension $n + 1$ , with ladder action of the standard generators $H$ , $E$ , $F$ . The general theory of weight decompositions, highest-weight vectors, and complete reducibility all originate as direct generalisations of the $sl_{2}$ ladder construction we ran in this unit. Every higher-rank semisimple Lie algebra contains $sl_{2}$ -triples indexed by its roots, and the restriction of any module along each such triple is a direct sum of $V_{n}$ 's.
Root system 07.06.03. The root-system description of a semisimple Lie algebra $g$ assigns to each root $α$ a copy of $sl_{2}$ inside $g$ (the $sl_{2}$ -triple ${H_{α}, E_{α}, F_{α}}$ ), and the highest-weight theorem for $g$ -representations reduces to the highest-weight theorem for each of these $sl_{2}$ 's. The Weyl-group reflection $s_{α}$ on the weight lattice is the rank-one symmetry $H \mapsto - H$ that swaps the top and bottom of the $sl_{2}$ ladder. The integrality of weights for $g$ -representations is exactly the integrality $n \in Z_{\geq 0}$ of $V_{n}$ propagated along each root.
Verma module 07.06.06. The Verma module $M (n)$ of $sl_{2} C$ is the infinite-dimensional highest-weight module generated by a free chain $v_{n}, F v_{n}, F^{2} v_{n}, \dots$ with no terminating relation. The finite-dimensional irreducible $V_{n}$ is the unique irreducible quotient of $M (n)$ , obtained by quotienting the kernel generated by $F^{n + 1} v_{n}$ . The Verma-module construction generalises to arbitrary semisimple Lie algebras and is the universal highest-weight module, of which $V_{n}$ is the finite-dimensional quotient in the integrable case.
Weyl character formula 07.06.07. The character of $V_{n}$ is $χ_{n} (t) = t^{n} + t^{n - 2} + \dots + t^{- n} = (t^{n + 1} - t^{- (n + 1)}) / (t - t^{- 1})$ , which is the Weyl character formula specialised to rank one. The Weyl numerator $t^{n + 1} - t^{- (n + 1)}$ is an alternating sum over the rank-one Weyl group $Z /2$ , and the Weyl denominator $t - t^{- 1}$ is the same alternating sum applied to the zero weight shifted by the Weyl vector $ρ = 1$ . The Weyl dimension formula reduces in rank one to $dim V_{n} = n + 1$ , exactly the count of rungs of the ladder.
Borel-Weil theorem 07.06.09. The Borel-Weil theorem realises every finite-dimensional irreducible representation of a compact Lie group as the space of holomorphic sections of a line bundle on a flag variety. For $sl_{2} C$ the flag variety is $C P^{1} = SU (2) / T$ , the line bundles are $O (n)$ for $n \in Z$ , and the global sections $H^{0} (C P^{1}, O (n))$ for $n \geq 0$ are exactly $V_{n}$ realised as homogeneous polynomials of degree $n$ in two variables. This is the geometric incarnation of the symmetric-power realisation.
Universal enveloping algebra 07.06.02. The Casimir element $C = E F + F E + H^{2} /2 \in U (sl_{2} C)$ is the rank-one instance of the quadratic Casimir of a semisimple Lie algebra. Its centrality in $U (sl_{2} C)$ powers the complete-reducibility theorem (Casimir separates irreducibles by eigenvalue) and the Verma-module decomposition (Casimir acts as a scalar on each weight space). The Harish-Chandra centre of $U (sl_{2} C)$ is the polynomial ring $C [C]$ , identified by the Harish-Chandra isomorphism with Weyl-invariants $C [H]^{Z /2} = C [H^{2}]$ .
Casimir element 07.06.10. Direct anchor. The dedicated Casimir unit develops the general construction $C = X_{a} X^{a} \in U (g)$ in arbitrary rank, the centrality proof, the eigenvalue formula $⟨ λ, λ + 2 ρ ⟩$ on $V_{λ}$ , and the Casimir-based proof of Weyl complete reducibility (the operator $(C - c_{2}) / (c_{1} - c_{2})$ splits short exact sequences with distinct eigenvalues; equal eigenvalues are handled by the Casimir-based vanishing of $Ext_{g}^{1} (V, V)$ ). The present unit is the canonical worked example of that machinery: $sl_{2} C$ has Casimir $C = E F + F E + H^{2} /2$ acting on $V_{n}$ by the scalar $n (n + 2) /2 = ⟨ n, n + 2 ⟩ /2$ , and the complete-reducibility proof for $sl_{2} C$ specialises the Casimir-eigenvalue separation to the rank-one ladder. Every Casimir computation in this unit — the centrality, the highest-weight evaluation, the separation of $V_{m}$ from $V_{n}$ for $m \neq = n$ — is a rank-one instance of the general construction in 07.06.10. The rescaling $ℓ (ℓ + 1) = j (j + 1)$ of the angular-momentum convention is the $su (2)$ specialisation of the Casimir eigenvalue $⟨ λ, λ + 2 ρ ⟩$ , foundational to quantum mechanics.
Representations of $sl_{3} C$ 07.06.12. The rank-two case is the direct successor: the $V_{a, b}$ classification of $sl_{3}$ irreducibles is built by applying the present $sl_{2}$ ladder construction along each of the two $sl_{2}$ -triples $(H_{α_{i}}, E_{α_{i}}, F_{α_{i}})$ in $sl_{3}$ ( $i = 1, 2$ ), with the integrality $a, b \in Z_{\geq 0}$ inherited from the rank-one integrality $n \in Z_{\geq 0}$ proved here. The hexagonal weight diagram of $V_{a, b}$ at 07.06.12 projects along any one simple-root direction onto a union of $sl_{2}$ -ladders of the present unit, recovering the rank-one picture as a slice of the rank-two picture. The propagation pattern — rank-one classification controls rank-two and beyond — recurs at every higher rank, with the $sl_{2}$ ladder of this unit serving as the universal local structure.

Historical & philosophical context [Master]

The classification of irreducible representations of $sl_{2} C$ emerged in stages across the late nineteenth and early twentieth centuries. Sophus Lie's 1880s work on continuous transformation groups introduced what we now call Lie algebras, and his collaborator Wilhelm Killing's 1888-1890 Mathematische Annalen papers Die Zusammensetzung der stetigen endlichen Transformations-Gruppen (vols 31, 33, 34, 36) classified the simple complex Lie algebras into the now-familiar series $A_{n}, B_{n}, C_{n}, D_{n}$ and the exceptions $E_{6}, E_{7}, E_{8}, F_{4}, G_{2}$ ^[pending]. Killing's classification was incomplete and contained errors; Élie Cartan's 1894 doctoral thesis Sur la structure des groupes de transformations finis et continus (Faculté des Sciences de Paris) ^[pending] gave the first complete and correct classification, with $sl_{2}$ (in Killing-Cartan notation $A_{1}$ ) as the rank-one base case. Cartan's thesis already contained the essential structure: $sl_{2}$ is the smallest simple Lie algebra, every simple Lie algebra of higher rank contains $sl_{2}$ -triples indexed by its roots, and the representation theory of $sl_{2}$ propagates to the higher-rank cases through these triples.

The representation theory itself was finalised by Hermann Weyl in his foundational 1925-26 papers Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen (Math. Z. 23, 271-309 and Math. Z. 24, 328-376, 377-395, 789-791) ^[pending]. Weyl proved the complete-reducibility theorem (every finite-dimensional representation of a semisimple Lie algebra is a direct sum of irreducibles) using what he called the unitarian trick: pass from the complex Lie algebra to its compact real form, exponentiate to a compact Lie group, average an arbitrary inner product to obtain an invariant one, and use orthogonal complementation to construct invariant decompositions. The same papers contained the Weyl character formula and the dimension formula, of which the $sl_{2}$ case is the rank-one specialisation. The name Casimir element honours Hendrik Casimir's 1931 thesis at Leiden under Paul Ehrenfest, in which the quadratic central element was identified as the natural Lie-algebraic invariant; the broader theory of central elements in the universal enveloping algebra was developed by Harish-Chandra in his 1951 Trans. AMS papers On some applications of the universal enveloping algebra of a semisimple Lie algebra ^[pending], establishing the Harish-Chandra isomorphism between the centre of $U (g)$ and the Weyl-invariants on the dual of the Cartan subalgebra.

The physics-mathematics dictionary places $sl_{2} C$ at the centre of quantum mechanics. The angular-momentum algebra of Wigner-Eckart and the spin algebra of Pauli are the same $sl_{2} C$ acting on Hilbert spaces of states, with $V_{n}$ the spin- $n /2$ representation. Eugene Wigner's 1939 Ann. Math. paper On unitary representations of the inhomogeneous Lorentz group identified elementary particles with irreducible unitary representations of the Poincaré group, and the $SU (2)$ -content of these representations (after the unitarian trick) is the spin classification: a single non-negative half-integer $j$ per particle, with $2 j + 1 = dim V_{n}$ internal states. The ladder operators $E$ and $F$ recur throughout quantum mechanics as creation and annihilation operators, the Casimir $C = J^{2}$ as the total angular momentum squared, and the eigenvalue $n (n + 2) /2 = j (j + 1) - 1/2$ as the (rescaled) total angular momentum quantum number.

Bibliography [Master]

@phdthesis{Cartan1894Thesis,
  author    = {Cartan, {\'E}lie},
  title     = {Sur la structure des groupes de transformations finis et continus},
  school    = {Facult{\'e} des Sciences de Paris},
  year      = {1894}
}

@article{Killing1888,
  author  = {Killing, Wilhelm},
  title   = {Die Zusammensetzung der stetigen endlichen Transformations-Gruppen},
  journal = {Mathematische Annalen},
  volume  = {31--36},
  year    = {1888--1890}
}

@article{Weyl1925a,
  author  = {Weyl, Hermann},
  title   = {Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. I},
  journal = {Mathematische Zeitschrift},
  volume  = {23},
  year    = {1925},
  pages   = {271--309}
}

@article{Weyl1925b,
  author  = {Weyl, Hermann},
  title   = {Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. II, III, IV},
  journal = {Mathematische Zeitschrift},
  volume  = {24},
  year    = {1925--1926},
  pages   = {328--376, 377--395, 789--791}
}

@phdthesis{Casimir1931,
  author    = {Casimir, Hendrik B. G.},
  title     = {Rotation of a Rigid Body in Quantum Mechanics},
  school    = {Leiden University},
  year      = {1931}
}

@article{HarishChandra1951,
  author  = {Harish-Chandra},
  title   = {On some applications of the universal enveloping algebra of a semisimple {L}ie algebra},
  journal = {Trans. Amer. Math. Soc.},
  volume  = {70},
  year    = {1951},
  pages   = {28--96}
}

@article{Wigner1939,
  author  = {Wigner, Eugene P.},
  title   = {On unitary representations of the inhomogeneous {L}orentz group},
  journal = {Ann. of Math. (2)},
  volume  = {40},
  year    = {1939},
  pages   = {149--204}
}

@book{FultonHarris,
  author    = {Fulton, William and Harris, Joe},
  title     = {Representation Theory: A First Course},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {129},
  year      = {1991}
}

@book{Humphreys,
  author    = {Humphreys, James E.},
  title     = {Introduction to {L}ie Algebras and Representation Theory},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {9},
  year      = {1972}
}

@book{Serre,
  author    = {Serre, Jean-Pierre},
  title     = {Complex Semisimple {L}ie Algebras},
  publisher = {Springer-Verlag},
  year      = {1987}
}

@book{HallLie,
  author    = {Hall, Brian C.},
  title     = {{L}ie Groups, {L}ie Algebras, and Representations: An Elementary Introduction},
  publisher = {Springer},
  edition   = {2},
  series    = {Graduate Texts in Mathematics},
  volume    = {222},
  year      = {2015}
}

@book{Kac,
  author    = {Kac, Victor G.},
  title     = {Infinite Dimensional {L}ie Algebras},
  publisher = {Cambridge University Press},
  edition   = {3},
  year      = {1990}
}

@article{Kashiwara1991,
  author  = {Kashiwara, Masaki},
  title   = {On crystal bases of the $Q$-analogue of universal enveloping algebras},
  journal = {Duke Math. J.},
  volume  = {63},
  year    = {1991},
  pages   = {465--516}
}

Prerequisites

07.06.01
07.06.03

Tier anchors

beginner: Fulton-Harris §11 ladder-operator opening; Hall *Lie Groups, Lie Algebras, and Representations* Ch. 4 informal account of the spin-$n/2$ tower
intermediate: Fulton-Harris *Representation Theory* §11; Humphreys *Introduction to Lie Algebras and Representation Theory* §7; Serre *Complex Semisimple Lie Algebras* §IV
master: Cartan 1894 *Sur la structure des groupes de transformations finis et continus* (Thèse, Paris) — classification of simple Lie algebras with $\mathfrak{sl}_2$ as the base case; Weyl 1925-26 *Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen* (Math. Z. 23, 271–309; 24, 328–376, 377–395, 789–791) — unitarian trick and complete reducibility; Fulton-Harris *Representation Theory* §11 and §12 (ladder construction, Casimir, complete reducibility); Humphreys §7; Serre *Complex Semisimple Lie Algebras* §IV; Kac *Infinite Dimensional Lie Algebras* §1 (where the same ladder structure recurs at the level of affine Kac-Moody algebras)

References

TODO_REF
Fulton-Harris — Representation Theory · §11 Lecture 11 The representations of $\mathfrak{sl}_2 \mathbb{C}$; §12 Complete reducibility and the Casimir element
TODO_REF
Humphreys — Introduction to Lie Algebras and Representation Theory · §7 Representations of $\mathfrak{sl}(2, F)$; §6.2 Casimir element; §6.3 Weyl's theorem on complete reducibility
TODO_REF
Serre — Complex Semisimple Lie Algebras · §IV Linear representations of $\mathfrak{sl}_2$; the Casimir scalar and the proof of complete reducibility from the unitarian trick
TODO_REF
Cartan — Sur la structure des groupes de transformations finis et continus · Thèse, Faculté des Sciences de Paris (1894), Ch. VIII; classification of simple Lie algebras over $\mathbb{C}$ with $\mathfrak{sl}_2$ as the rank-one base case
TODO_REF
Weyl — Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen · Math. Z. 23 (1925) 271-309 and Math. Z. 24 (1925-26) 328-376, 377-395, 789-791; the unitarian-trick proof of complete reducibility and the character formula
TODO_REF
Hall — Lie Groups, Lie Algebras, and Representations · Ch. 4 Representations of $\mathfrak{sl}_2(\mathbb{C})$; ladder-operator construction and the Casimir element
TODO_REF
Kirillov — An Introduction to Lie Groups and Lie Algebras · §4 Representations of $\mathfrak{sl}(2, \mathbb{C})$; weight-space decomposition and the highest-weight theorem

Reviewer

TBD

Estimated time

beginner: 20m
intermediate: 45m
master: 80m