Representations of SL₂

Core Idea

The representation theory of SL₂(ℂ) (or equivalently its Lie algebra 𝔰𝔩₂(ℂ)) is the fundamental example in Lie theory. The Lie algebra 𝔰𝔩₂ is spanned by three elements e, f, h with [h,e] = 2e, [h,f] = −2f, [e,f] = h. The finite-dimensional irreducible representations are classified by a single non-negative integer n: for each n ≥ 0, there is a unique (n+1)-dimensional irreducible representation V(n) with highest weight n. The representation V(n) has a basis of weight vectors v_n, v_{n−2}, …, v_{−n} on which e raises weight by 2, f lowers weight by 2, and h acts by the weight.

Explainer

The Lie algebra 𝔰𝔩₂(ℂ) consists of 2×2 traceless complex matrices. It is 3-dimensional with standard basis: e = [[0,1],[0,0]] (strictly upper triangular), f = [[0,0],[1,0]] (strictly lower triangular), and h = [[1,0],[0,−1]] (diagonal). The commutation relations are [h,e] = 2e, [h,f] = −2f, [e,f] = h. These relations, not the specific matrices, determine the representation theory. The element h generates a Cartan subalgebra, e is a raising operator, and f is a lowering operator.

A representation V of 𝔰𝔩₂ decomposes into weight spaces V = ⊕_λ V_λ, where V_λ = {v ∈ V : h·v = λv}. The commutation relations force e to raise weights by 2 (if v ∈ V_λ, then e·v ∈ V_{λ+2}) and f to lower weights by 2. In a finite-dimensional representation, there must be a highest weight vector v_λ with e·v_λ = 0 (since weights are bounded above). Starting from v_λ and applying f repeatedly generates v_λ, f·v_λ, f²·v_λ, … until reaching the lowest weight. If the highest weight is n, this chain has length n+1, producing an (n+1)-dimensional space with weights n, n−2, …, −n.

The classification theorem states: for each integer n ≥ 0, there is a unique irreducible representation V(n) of dimension n+1, and every finite-dimensional representation is a direct sum of these. The proof uses the Casimir element C = h² + 2h + 4fe ∈ U(𝔰𝔩₂), which lies in the center of the universal enveloping algebra and acts as the scalar n(n+2) on V(n). Since this scalar is distinct for each n, the Casimir separates irreducibles. Complete reducibility follows from the fact that SU(2) (the compact real form) is compact, so the analogue of Maschke's theorem applies.

The Clebsch-Gordan formula describes tensor products: V(m) ⊗ V(n) ≅ V(m+n) ⊕ V(m+n−2) ⊕ ··· ⊕ V(|m−n|), a multiplicity-free direct sum. This is the mathematical content of angular momentum addition in quantum mechanics (with V(n) corresponding to spin n/2). The formula can be proved by comparing characters or by explicitly constructing highest weight vectors in the tensor product. The entire structure — weight space decomposition, highest weight classification, Casimir element, Clebsch-Gordan decomposition — generalizes to all semisimple Lie algebras, with 𝔰𝔩₂ providing the blueprint for the general theory via the root system.