Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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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.

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The Harish-Chandra map on the center is injective

Statement

If a central element z∈Z(U(g)) has zero Harish-Chandra projection, then z=0. Equivalently, the Harish-Chandra map on the center is injective.

Facts & Assumptions

Given: A central element z∈Z(U(g)) with pr⁡(z)=0.

[F1]

The shifted Harish-Chandra map is an algebra isomorphism (Harish-Chandra isomorphism for the center).

Proof

technique · direct
1.1

By [F1], the shifted Harish-Chandra map HC⁡ρ(w)(λ)=pr⁡(w)(λ−ρ) is an algebra isomorphism from Z(U(g)) to S(h)W. Therefore it is injective.

F1given
2.1

If pr⁡(z)=0, then HC⁡ρ(z)=0 as a polynomial function on h∗. Injectivity from step 1.1 then forces z=0.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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