LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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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.
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Odd self-Jacobi follows over the complex numbers

Statement

For a complex Lie superalgebra $g$ , every odd $x \in g_{\overset{ˉ}{1}}$ satisfies $[x, [x, x]] = 0$ .

Facts & Assumptions

Given: A complex Lie superalgebra $g$ and an odd element $x \in g_{\overset{ˉ}{1}}$ .

[F1]

Lie superalgebra over the complex numbers: for homogeneous elements, $[x, [y, z]] = [[x, y], z] + (- 1)^{∣ x ∣ ∣ y ∣} [y, [x, z]]$ , and $[a, b] = - (- 1)^{∣ a ∣ ∣ b ∣} [b, a]$ .

Proof

1.1givenF1algebra

Put $A = [x, [x, x]]$ . The signed Jacobi identity in [F1], with $y = z = x$ , gives $A = [[x, x], x] - A$ , because $(- 1)^{∣ x ∣ ∣ x ∣} = - 1$ .

2.1givenF1step 1.1algebra∎

The element $[x, x]$ is even, so graded skew symmetry in [F1] gives $[[x, x], x] = - [x, [x, x]] = - A$ . Substituting into step 1.1 yields $A = - 2 A$ , hence $3 A = 0$ ; since the scalar field is $C$ , $A = 0$ .

Depends on

Lie superalgebra over the complex numbers

Used by

Ordered monomial basis for the enveloping superalgebra Theorem

Dependency tree · two levels

3 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

Sigiswald Barbier, A minimal representation for the orthosymplectic Lie-superalgebra, §2.2.1 (standard reference, not scraped)
Lauren Caston and Rita Fioresi, Mathematical Foundations of Supersymmetry, §1.2 (standard reference, not scraped)