Representations of Semisimple Lie Algebras

⚡ Key Takeaways

One-line statement: The finite-dimensional representations of a complex semisimple Lie algebra are completely reducible and classified by their dominant integral highest weights. Key result in words: Every finite-dimensional module decomposes into a direct sum of irreducibles, and irreducibles correspond bijectively to dominant integral weights via the highest-weight theorem; characters and dimensions are computed by the Weyl character and dimension formulas. Where it applies: Lie theory, algebraic groups, quantum mechanics (angular momentum), the Standard Model of particle physics, geometric representation theory, integrable systems. Who/when: Élie Cartan (classification, 1894–1913), Wilhelm Killing (root systems, 1888–1890), Hermann Weyl (character formula, complete reducibility, 1925–1926), Harish-Chandra, Chevalley, Serre (mid–20th century).

💡 The Core Insight

Choose a Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$ — a maximal abelian subalgebra of simultaneously diagonalizable elements. Because the elements of $\mathfrak{h}$ act semisimply and commute, any representation $V$ decomposes into simultaneous eigenspaces (weight spaces) under $\mathfrak{h}$ . The remaining generators (root vectors) act as raising and lowering operators that shift between weight spaces. An irreducible module is generated from a single "highest" weight vector by repeated lowering, and its entire structure — which weights appear, with what multiplicities — is dictated by the symmetry of the Weyl group. Representation theory of $\mathfrak{g}$ thereby becomes the study of a lattice acted on by a finite reflection group.

DefinitionSemisimple Lie Algebra and the Cartan Decomposition

A finite-dimensional Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ is semisimple if its Killing form $\kappa(X,Y)=\operatorname{tr}(\operatorname{ad}X\,\operatorname{ad}Y)$ is nondegenerate (Cartan's criterion). Fix a Cartan subalgebra $\mathfrak{h}$ . The adjoint action of $\mathfrak{h}$ decomposes $\mathfrak{g}$ into root spaces:

1

\quad \mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}_\alpha,\qquad \mathfrak{g}_\alpha=\{X\in\mathfrak{g}: [H,X]=\alpha(H)X\ \ \forall H\in\mathfrak{h}\}.

Derivation hint: Since $\operatorname{ad}\mathfrak{h}$ is a commuting family of semisimple operators on $\mathfrak{g}$ , it is simultaneously diagonalizable; the nonzero joint eigenvalues are the linear functionals $\alpha\in\mathfrak{h}^*$ called roots, with $\mathfrak{g}_0=\mathfrak{h}$ . Here $\Phi\subset\mathfrak{h}^*\setminus\{0\}$ is the root system, each $\dim\mathfrak{g}_\alpha=1$ . Choosing a set of positive roots $\Phi^+$ and simple roots $\Delta=\{\alpha_1,\dots,\alpha_\ell\}$ gives the triangular decomposition $\mathfrak{g}=\mathfrak{n}^-\oplus\mathfrak{h}\oplus\mathfrak{n}^+$ , where $\mathfrak{n}^\pm=\bigoplus_{\alpha\in\Phi^+}\mathfrak{g}_{\pm\alpha}$ .

DefinitionWeights, Dominant Integral Weights

For each root there is an $\mathfrak{sl}_2$ -triple $(e_\alpha,h_\alpha,f_\alpha)$ with coroot $h_\alpha=\alpha^\vee$ . Given a representation $V$ , a weight is $\lambda\in\mathfrak{h}^*$ with nonzero weight space

2

\quad V_\lambda=\{v\in V: H\cdot v=\lambda(H)v\ \ \forall H\in\mathfrak{h}\},\qquad V=\bigoplus_{\lambda} V_\lambda.

Derivation hint: Apply the simultaneous diagonalizability of the commuting semisimple operators $\rho(\mathfrak{h})$ , guaranteed because $\rho$ maps $\mathfrak{h}$ to commuting semisimple operators (a consequence of preservation of Jordan decomposition for semisimple $\mathfrak{g}$ ). The weight lattice $P=\{\lambda\in\mathfrak{h}^*:\langle\lambda,\alpha^\vee\rangle\in\mathbb{Z}\ \forall\alpha\in\Phi\}$ contains the dominant integral weights

3

\quad P^+=\{\lambda\in P: \langle\lambda,\alpha_i^\vee\rangle\geq 0,\ i=1,\dots,\ell\},\qquad \langle\lambda,\alpha^\vee\rangle=\frac{2(\lambda,\alpha)}{(\alpha,\alpha)}.

Derivation hint: Integrality follows from the representation theory of each $\mathfrak{sl}_2$ triple, where $h_\alpha$ acts with integer eigenvalues; dominance selects the highest-weight chamber under the Weyl group.

TheoremTheorem of the Highest Weight (Cartan–Weyl)

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. 1. Every finite-dimensional irreducible $\mathfrak{g}$ -module $V$ contains a unique (up to scalar) highest weight vector $v^+$ annihilated by $\mathfrak{n}^+$ , with weight $\lambda\in P^+$ , and $V$ is generated by $v^+$ under $\mathfrak{n}^-$ . 2. Two finite-dimensional irreducibles are isomorphic iff they have the same highest weight. 3. For every $\lambda\in P^+$ there exists a finite-dimensional irreducible $L(\lambda)$ with highest weight $\lambda$ . Thus there is a bijection

4

\quad \{\text{f.d. irreducible }\mathfrak{g}\text{-modules}\}/\!\cong\ \ \longleftrightarrow\ \ P^+.

Derivation hint: Existence comes from the Verma module $M(\lambda)=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\mathbb{C}_\lambda$ and its unique irreducible quotient $L(\lambda)$ ; finite-dimensionality of $L(\lambda)$ for dominant integral $\lambda$ follows from the $\mathfrak{sl}_2$ integrability conditions and Weyl-group stability of the weight set.

TheoremWeyl's Complete Reducibility

Every finite-dimensional representation $V$ of a semisimple Lie algebra $\mathfrak{g}$ is a direct sum of irreducible submodules:

5

\quad V\cong\bigoplus_{\lambda\in P^+} L(\lambda)^{\oplus m_\lambda},\qquad m_\lambda\in\mathbb{Z}_{\geq 0}.

Derivation hint: Use the Casimir element $\Omega$ (an invariant in the center of $U(\mathfrak{g})$ built from a basis dual under $\kappa$ ); $\Omega$ acts as a scalar $(\lambda,\lambda+2\delta)$ on $L(\lambda)$ , allowing extensions to be split — equivalently invoke vanishing of $H^1(\mathfrak{g},V)$ via Whitehead's lemma.

TheoremWeyl Character and Dimension Formulas

For $\lambda\in P^+$ , let $\delta=\tfrac12\sum_{\alpha\in\Phi^+}\alpha$ (the Weyl vector) and $W$ the Weyl group. The formal character $\operatorname{ch}L(\lambda)=\sum_\mu \dim L(\lambda)_\mu\, e^\mu$ satisfies

6

\quad \operatorname{ch}L(\lambda)=\frac{\sum_{w\in W}(-1)^{\ell(w)}\,e^{w(\lambda+\delta)}}{\sum_{w\in W}(-1)^{\ell(w)}\,e^{w(\delta)}}=\frac{\sum_{w\in W}(-1)^{\ell(w)}\,e^{w(\lambda+\delta)}}{\prod_{\alpha\in\Phi^+}\big(e^{\alpha/2}-e^{-\alpha/2}\big)}.

Derivation hint: Both numerator and denominator are $W$ -antiinvariant; using $\operatorname{ch}M(\lambda)=e^\lambda/\prod_{\alpha>0}(1-e^{-\alpha})$ and the Bernstein–Gelfand–Gelfand resolution of $L(\lambda)$ by Verma modules gives the alternating sum. Taking the limit $e^\mu\to e^{\langle\mu,t\rho\rangle}$ , $t\to0$ yields the Weyl dimension formula

7

\quad \dim L(\lambda)=\prod_{\alpha\in\Phi^+}\frac{(\lambda+\delta,\alpha)}{(\delta,\alpha)}.

Derivation hint: Apply L'Hôpital / the product expansion to [6] along the line through $\delta$ , where the Weyl denominator factorizes as the product over positive roots and antisymmetry produces the ratio of inner products.

ProofConstruction and Finiteness of

L(\lambda)

Step 1 — Verma module. Let $\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{n}^+$ . Form the one-dimensional $\mathfrak{b}$ -module $\mathbb{C}_\lambda$ where $\mathfrak{h}$ acts by $\lambda$ and $\mathfrak{n}^+$ acts by $0$ . Induce:

8

\quad M(\lambda)=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\mathbb{C}_\lambda.

Derivation hint: By the Poincaré–Birkhoff–Witt theorem, $M(\lambda)\cong U(\mathfrak{n}^-)\otimes\mathbb{C}_\lambda$ as a vector space, with a basis of ordered monomials in lowering operators applied to $v^+$ .

Step 2 — Unique irreducible quotient. $M(\lambda)$ has a unique maximal proper submodule (the sum of all submodules avoiding $V_\lambda$ ), so $L(\lambda)=M(\lambda)/\mathrm{(rad)}$ is irreducible with highest weight $\lambda$ .

Step 3 — Integrability. When $\langle\lambda,\alpha_i^\vee\rangle=n_i\in\mathbb{Z}_{\geq0}$ , the vector $f_i^{\,n_i+1}v^+$ is a singular (highest-weight) vector of weight $\lambda-(n_i+1)\alpha_i$ . Quotienting it out makes the action of each $\mathfrak{sl}_2$ -triple $(e_i,h_i,f_i)$ locally finite.

Step 4 — Local finiteness $\Rightarrow$ finite dimension. Local $\mathfrak{sl}_2$ -finiteness forces the set of weights of $L(\lambda)$ to be $W$ -invariant. A $W$ -invariant set of integral weights lying below $\lambda$ is finite (it sits inside the convex hull of $W\lambda$ intersected with the lattice), and each weight space is finite-dimensional. Hence $\dim L(\lambda)<\infty$ . $\blacksquare$

∎

ProofComplete Reducibility via the Casimir

Step 1. Choose a basis $\{X_i\}$ of $\mathfrak{g}$ and dual basis $\{X^i\}$ w.r.t. the Killing form $\kappa$ . The Casimir element

9

\quad \Omega=\sum_i X_i X^i\in Z(U(\mathfrak{g}))

Derivation hint: $\Omega$ is independent of the basis and commutes with all of $\mathfrak{g}$ because $\kappa$ is invariant ( $\kappa([Z,X],Y)+\kappa(X,[Z,Y])=0$ ); this is verified by a direct bracket computation.

Step 2. By Schur's lemma $\Omega$ acts on $L(\lambda)$ as the scalar $c_\lambda=(\lambda+\delta,\lambda+\delta)-(\delta,\delta)=(\lambda,\lambda+2\delta)$ .

Step 3. Suppose $0\to W\to V\to V/W\to 0$ is a short exact sequence of finite-dimensional modules. Reduce to the case where $W$ and $V/W$ are irreducible. If their Casimir scalars differ, $\Omega$ 's eigenspace decomposition splits the sequence. If they coincide, one handles the trivial-quotient case using $H^1(\mathfrak{g},\operatorname{Hom}(V/W,W))=0$ (Whitehead's first lemma), again provable through $\Omega$ .

Step 4. Iterating splits $V$ into a direct sum of irreducibles. $\blacksquare$

∎

Figure

The

A_2

(

\mathfrak{sl}_3

) root system with the dominant Weyl chamber and the weight diagram of the adjoint representation

```svg

```

HW

Hermann Weyl

1885–1955

Weyl unified the algebraic classification of Cartan with the analytic machinery of compact group integration. His unitarian trick transfers complete reducibility from compact Lie groups (where averaging over the group with Haar measure makes any representation unitary, hence completely reducible) to complex semisimple Lie algebras via their compact real forms. In 1925–26 he derived the character formula [6] and dimension formula [7], turning representation theory into explicit combinatorics. Weyl also coined much of the modern language (weights, the Weyl group, the Weyl chamber) and later applied these ideas to quantum mechanics, where the group $SU(2)$ governs angular momentum and gauge groups organize fundamental interactions.

🧠 Intuition

Think of $\mathfrak{h}$ as a set of commuting "energy meters." In any representation, vectors can be tuned to be simultaneous eigenvectors — these are the weight vectors, and their eigenvalues (weights) are points in a lattice. The root vectors $e_\alpha,f_\alpha$ act like the ladder operators of the harmonic oscillator or angular momentum: $e_\alpha$ raises a weight by $\alpha$ , $f_\alpha$ lowers it. An irreducible module is a single connected "crystal" of weights, all reachable from the topmost weight $\lambda$ by stepping down with lowering operators. The Weyl group $W$ is the symmetry group of this crystal — reflections in hyperplanes perpendicular to roots — so the weight diagram always looks the same from every chamber. Multiplicities (how many vectors share a weight) grow toward the center, exactly as the Weyl character formula's alternating sum predicts. The whole theory is the geometry of a lattice polytope decorated with integers.

Example

\mathfrak{sl}_2(\mathbb{C})

: the spin-

j

representations

Given: $\mathfrak{g}=\mathfrak{sl}_2$ with basis $e=\begin{pmatrix}0&1\\0&0\end{pmatrix}$ , $f=\begin{pmatrix}0&0\\1&0\end{pmatrix}$ , $h=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ , relations $[h,e]=2e$ , $[h,f]=-2f$ , $[e,f]=h$ . There is one simple root $\alpha$ with $\alpha^\vee=h$ ; $P=\mathbb{Z}\omega$ with $\langle\omega,h\rangle=1$ .

Steps. 1. Dominant integral weights: $\lambda=n\omega$ , $n\in\mathbb{Z}_{\geq0}$ . Highest weight $\lambda(h)=n$ . 2. Build $L(n)$ : highest weight vector $v_0$ with $h v_0=n v_0$ , $e v_0=0$ . Apply $f$ : vectors $v_k=f^k v_0$ have weight $n-2k$ , $k=0,\dots,n$ . 3. Action: $h v_k=(n-2k)v_k$ , $f v_k=v_{k+1}$ , $e v_k=k(n-k+1)v_{k-1}$ . The chain terminates: $f v_n=0$ . 4. Dimension formula [7]: only positive root $\alpha$ , $\delta=\omega$ , $(\lambda+\delta,\alpha)/(\delta,\alpha)=(n+1)/1$ .

Boxed answer:

\boxed{\dim L(n\omega)=n+1,\quad \text{weights }=\{n,n-2,\dots,-n\}.}

In physics this is the spin $j=n/2$ representation, dimension $2j+1$ (dimensionless count of states).

Example

\mathfrak{sl}_3(\mathbb{C})

: dimension of the irreducible

L(\omega_1+\omega_2)

(the adjoint)

Given: $\mathfrak{g}=\mathfrak{sl}_3$ , simple roots $\alpha_1,\alpha_2$ with $(\alpha_i,\alpha_i)=2$ , $(\alpha_1,\alpha_2)=-1$ . Positive roots: $\alpha_1,\ \alpha_2,\ \alpha_1+\alpha_2$ . Fundamental weights $\omega_1,\omega_2$ with $\langle\omega_i,\alpha_j^\vee\rangle=\delta_{ij}$ . Weyl vector $\delta=\omega_1+\omega_2$ .

Steps. Take $\lambda=\omega_1+\omega_2$ , so $\lambda+\delta=2\omega_1+2\omega_2$ . Compute the three factors of [7], using $(\mu,\alpha)=\tfrac12\langle\mu,\alpha^\vee\rangle(\alpha,\alpha)=\langle\mu,\alpha^\vee\rangle$ since $(\alpha,\alpha)=2$ .

root $\alpha$	$\langle\delta,\alpha^\vee\rangle$	$\langle\lambda+\delta,\alpha^\vee\rangle$
$\alpha_1$	$1$	$2$
$\alpha_2$	$1$	$2$
$\alpha_1+\alpha_2$	$2$	$4$

(For $\alpha_1+\alpha_2$ : $\langle\delta,(\alpha_1+\alpha_2)^\vee\rangle = \langle\omega_1+\omega_2,\alpha_1^\vee+\alpha_2^\vee\rangle=2$ .)

\dim L(\lambda)=\frac{2}{1}\cdot\frac{2}{1}\cdot\frac{4}{2}=8.

Boxed answer:

\boxed{\dim L(\omega_1+\omega_2)=8.}

This is the 8-dimensional adjoint representation of $\mathfrak{sl}_3$ — the "eightfold way" octet of the Gell-Mann classification of light hadrons (dimensionless multiplicity = 8 states).

⚠️ Common mistake

Complete reducibility fails without semisimplicity. Weyl's theorem [5] requires $\mathfrak{g}$ semisimple. For a non-semisimple algebra (e.g., one with a nontrivial abelian ideal) representations can be indecomposable but reducible. Also note that complete reducibility is a finite-dimensional statement: infinite-dimensional modules (Verma modules) are reducible but generally not completely reducible. Finally, the bijection [4] is to dominant integral weights — arbitrary $\lambda\in\mathfrak{h}^*$ give infinite-dimensional $L(\lambda)$ .

Counterexample

Consider $\mathfrak{g}=\mathfrak{gl}_2$ acting naturally, or more simply the 2-dimensional non-abelian solvable algebra $\mathfrak{b}$ with basis $h,e$ , $[h,e]=e$ . Its 2-dimensional representation

h\mapsto\begin{pmatrix}1&0\\0&0\end{pmatrix},\quad e\mapsto\begin{pmatrix}0&1\\0&0\end{pmatrix}

has the invariant line $\langle e_1\rangle$ but no complementary invariant line, since any $h$ -eigenvector for eigenvalue $0$ is mapped by $e$ into $\langle e_1\rangle$ . Thus the representation is reducible but not decomposable — exhibiting the failure of [5] outside the semisimple world.

connection

The classification rests on Root Systems and their Cartan matrices, encoded combinatorially by Dynkin Diagrams ( $A_n,B_n,C_n,D_n,E_6,E_7,E_8,F_4,G_2$ ). The reflection symmetry of weight diagrams is the action of the Weyl Group, a finite Coxeter group. Construction of $L(\lambda)$ goes through the Universal Enveloping Algebra and Verma Modules, whose composition series is governed by the BGG resolution and ultimately by Kazhdan–Lusztig Theory. By the Lie group–Lie algebra correspondence, these results pass to Compact Lie Groups (the Peter–Weyl theorem) and to algebraic groups. The character formula is a shadow of the Borel–Weil–Bott Theorem, realizing $L(\lambda)$ as sheaf cohomology of line bundles on the flag variety $G/B$ .

🔬 Where this appears

⏳ History

- 1888–1890 — Wilhelm Killing classifies simple Lie algebras and discovers root systems, though with gaps and errors.

- 1894 — Élie Cartan's thesis rigorously classifies complex simple Lie algebras, fixing Killing's work and identifying the exceptional types.

- 1913 — Cartan proves the highest-weight classification of irreducible finite-dimensional representations, theorem [4].

- 1925 — Weyl introduces the unitarian trick, proving complete reducibility via compact real forms.

- 1925–26 — Weyl derives the character formula [6] and dimension formula [7]; Issai Schur contributes parallel work on

GL_n

.

- 1937 — Harish-Chandra and others develop the algebraic (characteristic-zero) proofs avoiding analysis.

- 1948–1955 — Chevalley constructs integral forms, enabling Lie theory over arbitrary fields; the Chevalley basis.

- 1966 — Serre gives the presentation of

\mathfrak{g}

by generators and the Serre relations from the Cartan matrix.

- 1971–1976 — Bernstein–Gelfand–Gelfand build the BGG resolution and category

\mathcal{O}

, recovering [6] homologically and setting the stage for Kazhdan–Lusztig theory (1979).

open-problem

In positive characteristic, complete reducibility fails and the irreducible representations of reductive algebraic groups are vastly more intricate; Lusztig's conjecture on their characters (proven for large primes, false for small primes by Williamson's counterexamples, 2013) shows the combinatorics governed by $p$ -canonical bases is still not fully understood. The dimensions and characters of simple modules in the Lusztig small-quantum-group and modular settings remain open in general. For infinite-dimensional highest-weight modules, the determination of composition multiplicities of Verma modules in category $\mathcal{O}$ is the content of the (now-proven) Kazhdan–Lusztig conjecture, but analogous problems for Kac–Moody algebras, affine Lie algebras at non-generic levels, and W-algebras continue to drive research. Even computing tensor-product (fusion) multiplicities efficiently — generalized Littlewood–Richardson coefficients — is computationally hard for exceptional types.

Representations of Semisimple Lie Algebras

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also