Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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  • 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.
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A Verma composition factor has the same central character

Statement

If [M(λ):L(μ)]≠0, then χλ=χμ.

Facts & Assumptions

Given: Central characters Central character of a Lie algebra module, their scalar action Central elements act by scalars on cyclic highest-weight modules, and the simple quotient notation A Verma module has a unique simple quotient.

Proof

technique · direct
1.1

Every z∈Z(U(g)) acts on the cyclic highest-weight module M(λ) by the scalar χλ(z).

given
2.1

The same scalar action descends to each subquotient; on the composition factor L(μ) it is by definition χμ(z). Hence the two characters agree on every z.

step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

7 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