ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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The affine a1 gcm has singular rank one realization data

Example

The GCM $A = (\begin{array}{cc} 2 & - 2 \\ - 2 & 2 \end{array})$ has rank one and requires a three-dimensional minimal Cartan. On a basis $h_{0}, h_{1}, d$ , take simple-root coordinates $α_{0} = (2, - 2, 1)$ and $α_{1} = (- 2, 2, 0)$ .

Facts & Assumptions

Given: The displayed matrix and independent h_0,h_1,d.

[F1]

The smallest Cartan dimension with both families independent is 2n−rank A. (Minimal realizations exist and are unique up to isomorphism).

Verification

1.1

The second matrix row is the negative of the first, and the first is nonzero; hence $rank A = 1$ . Its diagonal is 2 and both off-diagonal entries are −2, so it is a GCM. Here $n = 2$ , and F1 gives minimal dimension $4 - 1 = 3$ .

F1given

2.1

The coordinate values give $α_{0} (h_{0}) = 2$ , $α_{1} (h_{0}) = - 2$ , $α_{0} (h_{1}) = - 2$ , $α_{1} (h_{1}) = 2$ . If $a α_{0} + b α_{1} = 0$ , evaluation on $d$ gives $a = 0$ , then on $h_{0}$ gives $b = 0$ . Thus the two roots, as well as the supplied two coroots, are independent. Both roots annihilate the nonzero vector $h_{0} + h_{1}$ , while $(α_{0} + α_{1}) (d) = 1$ . Omitting $d$ would make the roots negatives of one another and destroy independence. The data are therefore a minimal realization with an essential complementary direction.

F1step 1.1∎

Sources

Source comparison: Kleshchev, Example 1.2.3 and Proposition 1.2.4, pp.10–12; affine A1 data in Example 1.5.4, pp.23–24.

Depends on

Minimal realizations exist and are unique up to isomorphism

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Example 1.2.3 and Proposition 1.2.4, pp.10–12; affine A1 data in Example 1.5.4, pp.23–24 (standard reference, not scraped)