False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedaudited 2026-09-22

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Root systems determine only isogeny class

Statement

Assume the Axiom of Choice. A root system determines a compact connected semisimple group up to isomorphism.

Facts & Assumptions

Given: Assume the Axiom of Choice; the groups $S U (2)$ and $S O (3)$ with their standard structures.

[L1]

The simply connected form and the adjoint form of a root system are centrally isogenous but generally not isomorphic; for type $A_{1}$ the simply connected form is $S U (2)$ and the adjoint form is $S O (3) = S U (2) / {\pm I}$ (Semisimple compact groups up to isogeny, Central quotients and intermediate character lattices).

[L2]

The centre of $S U (2)$ is ${\pm I}$ , of order two, while the centre of $S O (3)$ is trivial: a central element of $S O (3)$ commutes with every rotation, and a rotation commuting with all rotations is the identity. [L1]

Refutation

technique · direct

1.1

Both $S U (2)$ and $S O (3)$ are compact, connected and semisimple, and both have root system of type $A_{1}$ ; indeed $S O (3)$ is the quotient of $S U (2)$ by the central subgroup ${\pm I}$ of order two by [L1], and the quotient map is a finite central isogeny.

1.2

An isomorphism of Lie groups carries the centre onto the centre; by [L2] the centres are ${\pm I}$ for $S U (2)$ and the trivial group for $S O (3)$ , so no isomorphism exists.

2.1

Hence the type $A_{1}$ root system determines $S U (2)$ and $S O (3)$ only up to finite central isogeny, not up to isomorphism; the statement is false and the correct classification theorem retains the isogeny qualification, with the intermediate central quotients recorded by their character lattices.

L1step 1.1step 1.2∎

Depends on

Used by

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Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)