Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-07
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Verma homomorphisms and singular vectors

Statement

For weights λ,μ, evaluation at the highest-weight vector gives a natural vector-space isomorphism Hom⁡g(M(μ),M(λ))≅{v∈M(λ)μ:n+v=0}.

Facts & Assumptions

Given: The universal property of The universal property of Verma modules.

Proof

technique · direct
1.1

A homomorphism f sends the highest-weight vector vμ to a vector of weight μ killed by n+; evaluation is therefore a linear map into the displayed space.

given
2.1

Conversely, a vector v in that space is a highest-weight vector of weight μ, so the universal property supplies a unique homomorphism M(μ)→M(λ) sending vμ to v. The two constructions are inverse.

givenconstruct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources