Math 2.0

Nonassociative rings and algebras Representations of Lie algebras Theorem

Bernstein–Gelfand–Gelfand resolution

Builds on Semisimple Lie algebraWeyl groupVerma moduleTheorem of the highest weightExact sequence

Statement

Let 𝔤\mathfrak{g} be a complex semisimple Lie algebra with Cartan subalgebra 𝔥\mathfrak{h}, positive roots Φ+\Phi^+ and Weyl group WW. Let ρ\rho be half the sum of the positive roots, and let N=∣Φ+∣N = \abs{\Phi^+}. Let ℓ(w)\ell(w) be the length of w∈Ww \in W with respect to the simple reflections. For w∈Ww \in W and λ∈𝔥∗\lambda \in \mathfrak{h}^* put w⋅λ=w(λ+ρ)−ρw \cdot \lambda = w(\lambda + \rho) - \rho. Let λ\lambda be a dominant integral weight, and let L(λ)L(\lambda) be the finite-dimensional irreducible module of highest weight λ\lambda. Let M(μ)M(\mu) be the Verma module of highest weight μ\mu. Then there is an exact sequence of 𝔤\mathfrak{g}-modules 0→CN→⋯→C1→C0→L(λ)→0,Ck=⨁w∈W, ℓ(w)=kM(w⋅λ).0 \to C_N \to \dots \to C_1 \to C_0 \to L(\lambda) \to 0, \qquad C_k = \bigoplus_{w \in W,\ \ell(w) = k} M(w \cdot \lambda).

Remarks

The maps are built from embeddings of Verma modules. Bernstein, Gelfand and Gelfand showed that M(w⋅λ)M(w \cdot \lambda) embeds in M(w′⋅λ)M(w' \cdot \lambda) when w′≤ww' \le w in the Bruhat order. The components between consecutive terms are such embeddings, for ℓ(w)=ℓ(w′)+1\ell(w) = \ell(w') + 1, with signs chosen so that the composite of two maps is zero. Since each M(μ)M(\mu) is a free U(𝔫−)U(\mathfrak{n}^-)-module, the sequence is a free resolution of L(λ)L(\lambda) over U(𝔫−)U(\mathfrak{n}^-). Example: for 𝔰𝔩2(ℂ)\mathfrak{sl}_2(\C) and λ(h)=n≥0\lambda(h) = n \ge 0, it is 0→M(−n−2)→M(n)→L(n)→00 \to M(-n-2) \to M(n) \to L(n) \to 0. Taking characters, the alternating sum of the characters of the CkC_k gives the Weyl character formula.

Going deeper

For a student

Since the resolution is free over U(𝔫−)U(\mathfrak{n}^-), it computes the Lie algebra homology of 𝔫−\mathfrak{n}^- with coefficients in L(λ)L(\lambda). The result is Kostant's theorem: Hk(𝔫−,L(λ))H_k(\mathfrak{n}^-, L(\lambda)) is the sum of the one-dimensional 𝔥\mathfrak{h}-modules of weights w⋅λw \cdot \lambda with ℓ(w)=k\ell(w) = k. It is a basic tool in category O, where Verma modules are the standard objects. Variants for parabolic subalgebras use generalised Verma modules and the minimal length coset representatives of WW.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

Show the path as a graph
Cartan subalgebra, concept, no evidence levelCartan subalgebraconceptCommutative ring, concept, no evidence levelCommutative ringconceptCoroot, concept, no evidence levelCorootconceptHighest weight, concept, no evidence levelHighest weightconceptImage of a homomorphism, concept, no evidence levelImage of a homomorphi…conceptIrreducible Lie module, concept, no evidence levelIrreducible Lie moduleconceptKernel of a homomorphism, concept, no evidence levelKernel of a homomorph…conceptLinear map, concept, no evidence levelLinear mapconceptModule over a ring, concept, no evidence levelModule over a ringconceptSemisimple Lie algebra, concept, no evidence levelSemisimple Lie algebraconceptSimple roots, concept, no evidence levelSimple rootsconceptVerma module, concept, no evidence levelVerma moduleconceptWeyl group, concept, no evidence levelWeyl groupconceptTheorem of the highest weight, statement, no evidence levelTheorem of the highes…statementExact sequence, definition, no evidence levelExact sequencedefinitionBernstein–Gelfand–Gelfand resolution, statement, no evidence levelBernstein–Gelfand–Gel…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cartan subalgebra concept
  2. Commutative ring concept
  3. Coroot concept
  4. Highest weight concept
  5. Image of a homomorphism concept
  6. Irreducible Lie module concept
  7. Kernel of a homomorphism concept
  8. Linear map concept
  9. Module over a ring concept
  10. Semisimple Lie algebra concept
  11. Simple roots concept
  12. Verma module concept
  13. Weyl group concept
  14. Theorem of the highest weight statement
  15. Exact sequence definition
  16. Bernstein–Gelfand–Gelfand resolution statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Verma_module
About this page
Tag41KGWXZ0VM
URIm2://41KGWXZ0VM
Kindstatement, theorem
WritingW0 stub
MSC17B10, 17B55
StatusThis copy of the site has no build data. Run m2 status 41KGWXZ0VM.
AboutVerma module
UsesSemisimple Lie algebra, Weyl group, Verma module, Theorem of the highest weight, Exact sequence
Changed3 November 2026

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