LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Semisimple groups are perfect and have no nontrivial characters

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $G$ be a semisimple algebraic group over a field $k$ (Split reductive groups, Radical, unipotent radical, semisimple and reductive algebraic groups). Then $G = [G, G]$ (The derived subgroup, the derived series and solvable algebraic groups) and $X (G) = {Hom}_{k} (G, G_{m}) = 0$ ; equivalently, every one-dimensional rational representation of $G$ is trivial.

Facts & Assumptions

Given: A semisimple algebraic group $G$ over $k$ , its derived subgroup $[G, G] = G^{'}$ and its character group $X (G) = {Hom}_{k} (G, G_{m})$ .

[F1]

Semisimple groups are reductive. $R_{u} (G)$ is contained in the radical $R (G)$ and $G$ is semisimple exactly when $R (G_{k^{a}}) = 1$ , so $R_{u} (G_{k^{a}}) = 1$ and $G$ is reductive (Radical, unipotent radical, semisimple and reductive algebraic groups).

[F2]

Centre, radical and derived subgroup. For a reductive $G$ one has $G = Z (G)_{t} \cdot G^{'}$ with finite intersection, and $G^{'}$ is semisimple; moreover $G$ is semisimple if and only if $Z (G)$ is finite (Centre, radical and semisimple quotient of a reductive group, The derived subgroup, the derived series and solvable algebraic groups).

[F3]

Characters kill commutators. A morphism of algebraic groups $χ : G \to G_{m}$ satisfies $χ (g h g^{- 1} h^{- 1}) = 1$ for all $R$ -points $g, h$ , since $G_{m}$ is commutative; hence $χ$ is trivial on the derived subgroup $[G, G]$ (Properties of the derived subgroup of an algebraic group).

[F4]

One-dimensional representations are characters. A rational representation of $G$ on a one-dimensional $k$ -space is given by a morphism $G \to {GL}_{1} ≅ G_{m}$ , that is, by an element of $X (G)$ (Rational representations and comodules of an affine group scheme).

Proof

technique · direct

1.1F1F2given

By [F1] the group $G$ is reductive. Since $G$ is semisimple, [F2] shows that $Z (G)$ is finite, so its largest central torus $Z (G)_{t}$ is trivial, and the decomposition $G = Z (G)_{t} \cdot G^{'}$ of [F2] gives $G = G^{'} = [G, G]$ .

2.1F3step 1.1

Let $χ \in X (G)$ . By [F3] the character $χ$ is trivial on every commutator, hence on $[G, G]$ , which is all of $G$ by step 1.1; therefore $χ$ is the trivial character.

3.1F4step 2.1

By [F4] a one-dimensional rational representation of $G$ is given by a character on $G$ ; by step 2.1 every such representation is trivial.

4.1step 1.1step 3.1∎

Steps 1.1 and 3.1 give both $G = [G, G]$ and $X (G) = 0$ , and the displayed equivalence with triviality of all one-dimensional rational representations.

Remarks

No splitness is used; the argument applies to every semisimple algebraic group over $k$ .
The commutator identity used for characters is the only place where the commutativity of $G_{m}$ enters; it makes every character factor through the abelianization $G / [G, G]$ .

Depends on

Used by

Dependency tree · two levels

44 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

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
Robert Steinberg, Lectures on Chevalley Groups (Yale University, 1967; notes prepared by J. Faulkner and R. Wilson) (standard reference, not scraped)