ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generated

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.

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An odd abelian subalgebra has an exterior enveloping algebra

Example

Let $a = C e$ be the one-dimensional odd subalgebra spanned by $e$ inside the $g l (1 ∣ 1)$ of The gl(1|1) bracket data define a Lie superalgebra. Its bracket is zero because $[e, e] = 0$ . Then $U (a)$ is the two-dimensional exterior algebra $C \oplus C e$ , with $e^{2} = 0$ .

Facts & Assumptions

Given: The one-dimensional graded subspace $a = C e$ and the enveloping-superalgebra convention.

[F1]

Bracket data for gl(1|1): $e$ is odd and $[e, e] = 0$ .

[F2]

The gl(1|1) bracket data define a Lie superalgebra: the bracketed graded space is a Lie superalgebra.

[F3]

Universal enveloping superalgebra: the quotient is $T (a) / I$ , with the relation $ι_{a} (e)^{2} = ι_{a} ([e, e]) / 2$ for odd $e$ .

[F4]

Ordered monomial basis for the enveloping superalgebra: for an ordered homogeneous basis with one odd basis vector $e$ , the ordered monomials $1, e$ form a basis.

Verification

technique · direct

1.1F1F2F3algebra

By [F3] and [F1], the defining relation is $ι_{a} (e)^{2} = 0$ ; the tensor quotient is generated by $1$ and $ι_{a} (e)$ , so every word reduces to a linear combination of them.

2.1F3F4step 1.1∎

By [F4], $1, e$ are linearly independent in the quotient and span it, so $U (a) = C \oplus C e$ as a vector space with the stated multiplication.

Depends on

Used by

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Sources

Sigiswald Barbier, A minimal representation for the orthosymplectic Lie-superalgebra, §2.6.2 (standard reference, not scraped)