Representations of Semisimple Lie Algebras

A semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ is a finite-dimensional Lie algebra with trivial radical, equivalently a direct sum of simple Lie algebras. A representation is a Lie algebra homomorphism $\rho :\mathfrak{g}\to \mathfrak{gl}(V)$, making $V$ a $\mathfrak{g}$-module. The central miracle, Weyl's theorem on complete reducibility, says every finite-dimensional representation splits into irreducibles, so understanding the irreducibles understands everything. These irreducibles are governed entirely by the combinatorics of a root system and its weight lattice: each finite-dimensional irreducible has a unique highest weight, a dominant integral element of the dual Cartan subalgebra. The surprising fact is that a problem in infinite-dimensional analysis (operators on vector spaces) reduces to pure finite combinatorics — lattices, reflections, and a finite Weyl group — completely classifying all irreducible modules.

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.

Formal Statement

Semisimple Lie Algebra and the Cartan Decomposition

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

Here $\Phi \subset {\mathfrak{h}}^{\ast }\setminus \{0\}$ 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 ${\mathfrak{n}}^{\pm }={⨁}_{\alpha \in {\Phi }^{+}}{\mathfrak{g}}_{\pm \alpha }$.

Weights, 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 $\lambda \in {\mathfrak{h}}^{\ast }$ with nonzero weight space

The weight lattice $P=\{\lambda \in {\mathfrak{h}}^{\ast }:⟨\lambda ,{\alpha }^{\vee }⟩\in \mathbb{Z}\forall \alpha \in \Phi \}$ contains the dominant integral weights

Theorem 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

Weyl's Complete Reducibility

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

Weyl Character and Dimension Formulas

For $\lambda \in P^{+}$, let $\delta =\frac{1}{2}{\sum }_{\alpha \in {\Phi }^{+}}\alpha$ (the Weyl vector) and $W$ the Weyl group. The formal character $\mathrm{ch}L(\lambda )={\sum }_{\mu }\dim L(\lambda {)}_{\mu }{e}^{\mu }$ satisfies

Taking the limit $e^{\mu }\to e^{⟨\mu ,t\rho ⟩}$, $t\to 0$ yields the Weyl dimension formula

Why It Holds / Derivation

Construction and Finiteness of $L(\lambda)$

Complete Reducibility via the Casimir

The $A_2$ ($\mathfrak{sl}_3$) root system with the dominant Weyl chamber and the weight diagram of the adjoint representation

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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.

Building 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.

Worked Examples

$\mathfrak{sl}_2(\mathbb{C})$: the spin-$j$ representations

1. Dominant integral weights: $\lambda =n\omega$, $n\in {\mathbb{Z}}_{\ge 0}$. 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 $\alpha$, $\delta =\omega$, $(\lambda +\delta ,\alpha )/(\delta ,\alpha )=(n+1)/1$.

$\mathfrak{sl}_3(\mathbb{C})$: dimension of the irreducible $L(\omega_1+\omega_2)$ (the adjoint)

| root $\alpha$ | $⟨\delta ,{\alpha }^{\vee }⟩$ | $⟨\lambda +\delta ,{\alpha }^{\vee }⟩$ | |---|---|---| | ${\alpha }_1$ | $1$ | $2$ | | ${\alpha }_2$ | $1$ | $2$ | | ${\alpha }_1+{\alpha }_2$ | $2$ | $4$ |

(For ${\alpha }_1+{\alpha }_2$: $⟨\delta ,({\alpha }_1+{\alpha }_2{)}^{\vee }⟩=⟨{\omega }_1+{\omega }_2,{\alpha }_1^{\vee }+{\alpha }_2^{\vee }⟩=2$.)

Common Misconceptions

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

Connections

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 $G/B$.

Real-World Applications

1. Quantum angular momentum. The irreducibles of $\mathfrak{su}(2)\cong {\mathfrak{sl}}_2$ are the spin-$j$ multiplets; Clebsch–Gordan decomposition is the tensor-product reduction of these modules.
2. The Standard Model / Gell-Mann's eightfold way. $SU(3)$ flavor symmetry organizes hadrons into the $8$ and $10$ representations; the prediction of the ${\Omega }^{-}$ baryon came from completing the decuplet $L(3{\omega }_1)$.
3. Gauge theory. Yang–Mills fields take values in $\mathfrak{g}$; matter fields transform in representations $L(\lambda )$ of the gauge group (e.g. quarks in the fundamental $3$ of $SU(3{)}_{\text{color}}$).
4. Grand Unified Theories. $SU(5)$, $SO(10)$, and $E_6$ unification arrange Standard-Model fermions inside single irreducibles such as the $16$ of $SO(10)$.
5. Integrable systems & spin chains. The Bethe ansatz and quantum groups deform $U(\mathfrak{g})$; representation categories control the spectrum of the Heisenberg model.
6. Crystallography & molecular spectroscopy. Branching rules for restricting representations to subgroups predict spectral line splitting.
7. Geometric Langlands & automorphic forms. Satake isomorphism relates representations of the Langlands dual group $\hat{G}$ to Hecke algebras, a cornerstone of modern number theory.

Historical Development

• 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 $G{L}_n$.
• 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).

Subtleties & Limits

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.

See Also