ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedaudited 2026-09-22

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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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Simply connected, adjoint, and intermediate compact forms

Example

Assume the Axiom of Choice. For a semisimple compact root system, the simply connected form corresponds to the weight lattice $P$ , the adjoint form to the root lattice $Q$ , and intermediate finite central quotients to the intermediate lattices $Q \subseteq X \subseteq P$ .

Facts & Assumptions

Given: Assume the Axiom of Choice; a simply connected compact semisimple group $G_{s c}$ with maximal torus $T_{s c}$ , root lattice $Q$ and weight lattice $P$ .

[L1]

Central subgroups $C \leq Z (G_{s c})$ correspond bijectively and contravariantly to lattices $Q \subseteq X \subseteq P$ by $C \mapsto X^{*} (T_{s c} / C)$ ; the trivial central subgroup gives $P$ and the full centre gives $Q$ (Central quotients and intermediate character lattices).

[L2]

The simply connected compact form has character lattice $P$ and the adjoint form has character lattice $Q$ (Root and weight lattice sandwich).

Verification

technique · direct

1.1

By [L2] the simply connected form $G_{s c}$ itself has $X^{*} (T_{s c}) = P$ , corresponding under [L1] to the trivial central subgroup.

L1L2

1.2

The adjoint form $G_{a d} = G_{s c} / Z (G_{s c})$ has character lattice $Q$ by [L2]; by [L1] it corresponds to the full centre, and the annihilator of $Q$ in the finite dual pairing is exactly $Z (G_{s c})$ .

L1L2

2.1

For an intermediate lattice $Q ⊊ X ⊊ P$ the annihilator $C_{X} = ⋂_{χ \in X} \ker χ$ is a nontrivial proper central subgroup with $X^{*} (T_{s c} / C_{X}) = X$ , by [L1], and conversely every nontrivial proper central subgroup arises this way; so the intermediate quotients are exactly the intermediate lattices.

L1step 1.2

3.1

This yields the full menu of forms: the simply connected endpoint, the adjoint endpoint, and one marked quotient for each intermediate lattice, which is the sense in which compact semisimple groups are classified by root datum with the added lattice data.

L1step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

32 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

Brian Conrad and Aaron Landesman, Compact Lie Groups (standard reference, not scraped)