Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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Chevalley restriction for symmetric invariants

Statement

Let g be a complex semisimple Lie algebra, let h⊆g be a Cartan subalgebra, and let W be its Weyl group. Restriction to h induces an algebra isomorphism

S(g)g→∼S(h)W.

Facts & Assumptions

Given: A complex semisimple Lie algebra g, a Cartan subalgebra h, its Weyl group W, and the restriction map res⁡ ⁣:S(g)g→S(h)W.

Proof

technique · direct
1.1

The restriction map is injective by An invariant polynomial is determined by its restriction to a Cartan subalgebra.

given
1.2

The restriction map is surjective by Weyl-invariant polynomials on the Cartan extend to invariant polynomials on g, which constructs an adjoint-invariant extension for every Weyl-invariant polynomial on h.

given
2.1

Hence restriction is a bijective algebra homomorphism, so it is an algebra isomorphism.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

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Sources