Alphabeta Math
LemmaStatement: 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.
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An invariant polynomial is determined by its restriction to a Cartan subalgebra

Statement

Let g be a complex semisimple Lie algebra and let h⊆g be a Cartan subalgebra. If an adjoint-invariant polynomial on g vanishes on h, then it vanishes identically on g. Equivalently, an invariant polynomial is determined by its restriction to h.

Facts & Assumptions

Given: A complex semisimple Lie algebra g, a Cartan subalgebra h, and an adjoint-invariant polynomial function f∈S(g)g whose restriction to h is zero.

Proof

technique · direct
1.1

By invariance, if f vanishes on h, then it also vanishes on every conjugate of h.

given
2.1

By Regular semisimple elements form a dense open subset, every regular semisimple element is conjugate to an element of h, and those elements form a dense open subset of g. Hence f vanishes on a dense open subset of g.

step 1.1
3.1

A polynomial function on an irreducible affine space that vanishes on a dense subset is identically zero. Therefore f=0 on all of g.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

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Sources