ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A nonabelian group makes the double cross-relations noncommutative

Example

In $D (k S_{3})$ let $p = (12)$ , $q = (23)$ and $r = (13) = p q p^{- 1}$ . Then $(δ_{r} \otimes p)$ and $(δ_{q} \otimes e)$ do not commute. On the Yetter–Drinfeld module $k S_{3}$ with basis $u_{x}$ , degree $x$ and conjugation action, the braiding sends $u_{p} \otimes u_{q}$ to $u_{r} \otimes u_{p}$ , which differs from the ordinary flip.

Facts & Assumptions

Given: The group-double multiplication and Yetter–Drinfeld braiding, and the transpositions $p, q, r$ in $S_{3}$ . Verify the two products and the stated braiding value.

[L1]

The Drinfeld double of a finite group algebra: $(δ_{x} \otimes g) (δ_{y} \otimes h) = 1_{x = g y g^{- 1}} δ_{x} \otimes g h$ .

[L2]

Modules over the finite double are left-left Yetter–Drinfeld modules: $c (v \otimes w) = \sum v_{- 1} \cdot w \otimes v_{0}$ .

[A1]

In $S_{3}$ , $p q p^{- 1} = r = (13)$ , $q \neq r$ , and $p^{- 1} = p$ .

Verification

1.1givenA1L1algebra

The product formula [L1] gives $(δ_{r} \otimes p) (δ_{q} \otimes e) = δ_{r} \otimes p$ , since $r = p q p^{- 1}$ . Reversing the factors gives $(δ_{q} \otimes e) (δ_{r} \otimes p) = 0$ , since $q \neq r$ . Thus these elements do not commute.

2.1A1L2algebra∎

Define $u_{x}$ to have degree $x$ and let $g \cdot u_{x} = u_{g x g^{- 1}}$ . This action preserves the grading by conjugation, so it is a Yetter–Drinfeld module by the group case of [L2]. Its braiding is $c (u_{p} \otimes u_{q}) = u_{p q p^{- 1}} \otimes u_{p} = u_{r} \otimes u_{p}$ , whereas the ordinary flip gives $u_{q} \otimes u_{p}$ . These tensors differ because $r \neq q$ .

Depends on

Used by

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Dependency tree · two levels

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Sources

Kalle Kytölä, Introduction to Hopf algebras and representations, lecture notes, Exercise 42 (standard reference, not scraped)
G. A. García and C. Vay, Simple modules of small quantum groups at dihedral groups, Doc. Math. 29 (2024), §3.1.3 (standard reference, not scraped)