Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06
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A Verma module has a unique simple quotient

Statement

The proper submodule J(λ) which is the sum of all proper submodules is the unique maximal submodule of M(λ). The quotient L(λ):=M(λ)/J(λ) is simple and is its unique simple quotient.

Facts & Assumptions

Given: Properness of J(λ) from The sum of all proper Verma submodules is proper.

Proof

technique · direct
1.1

Every proper submodule is contained in J(λ) by its definition, so J(λ) is maximal and unique among proper maximal submodules.

givenalgebra
2.1

A submodule of M(λ)/J(λ) lifts to a submodule containing J(λ); it is either J(λ) or all of M(λ). Thus the quotient is simple, and the kernel of any simple quotient is a maximal submodule, necessarily J(λ).

givenalgebra∎

Depends on

Used by

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