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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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The regular Hopf module and its projection

Example

Let $H$ be any Hopf algebra over a field $k$ . Regard $H$ as a right module over itself by multiplication and as a right comodule by its coproduct $Δ$ . Its coinvariants are $k 1_{H}$ , and the projection from the fundamental theorem is $P (h) = ε (h) 1_{H}$ . The decomposition map $k 1_{H} \otimes H \to H$ is $λ 1_{H} \otimes h \mapsto λ h$ .

Facts & Assumptions

Given: A Hopf algebra $(H, m, 1_{H}, Δ, ε, S)$ over a field $k$ , with its regular right action and regular right coaction.

[F1]

In the right-right convention, compatibility is $ρ (m h) = \sum m_{0} h_{1} \otimes m_{1} h_{2}$ and coinvariants satisfy $ρ (m) = m \otimes 1_{H}$ (Right-right Hopf modules and coinvariants).

[F2]

The projection is $P (m) = \sum m_{0} S (m_{1})$ , and the fundamental theorem uses $α (n \otimes h) = n h$ and $β (m) = \sum P (m_{0}) \otimes m_{1}$ (The coinvariant projection of a right-right Hopf module, The fundamental theorem of right-right Hopf modules).

[F3]

The multiplication is unital and associative, $Δ$ is coassociative and a unital algebra map, the counit identities are $(ε \otimes id) Δ (h) = h = (id \otimes ε) Δ (h)$ under the tensor-unit identifications, and $\sum h_{1} S (h_{2}) = ε (h) 1_{H}$ (Hopf algebras and antipodes).

Verification

1.1F1F3algebra

Multiplication gives the regular right module action because it is associative and unital; $Δ$ gives the regular right coaction because it is coassociative and counital. They satisfy compatibility since $Δ (h g) = Δ (h) Δ (g) = \sum h_{1} g_{1} \otimes h_{2} g_{2}$ , which is exactly the right side of [F1] for $m = h$ .

2.1F2F3step 1.1∎

If $h$ is coinvariant, then $Δ (h) = h \otimes 1_{H}$ ; applying $ε \otimes id$ gives $h = ε (h) 1_{H}$ . Conversely every scalar multiple of $1_{H}$ is coinvariant. Moreover $P (h) = \sum h_{1} S (h_{2}) = ε (h) 1_{H}$ , so $β (h) = \sum ε (h_{1}) 1_{H} \otimes h_{2} = 1_{H} \otimes h$ and $α β (h) = h$ . These are precisely the stated coinvariant subspace and decomposition maps.

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