LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08

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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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Uniqueness of the antipode

Statement

Let $(A, m, η, Δ, ε)$ be a bialgebra over a commutative ring $R$ (Bialgebras, counits and antipodes over a commutative ring) and let $S, S^{'} : A \to A$ be antipodes. Then $S = S^{'}$ . Thus a bialgebra admits at most one Hopf algebra structure with its fixed multiplication, unit, coproduct and counit. Every antipode also satisfies $S (1_{A}) = 1_{A}$ .

Facts & Assumptions

Given: A bialgebra $(A, m, η, Δ, ε)$ over a commutative ring $R$ and two antipodes $S, S^{'}$ .

[F1]

The multiplication $m$ is associative and the coproduct $Δ$ is coassociative (Bialgebras, counits and antipodes over a commutative ring).

[F2]

The counit identities are $(ε \otimes id) Δ = id = (id \otimes ε) Δ$ (Bialgebras, counits and antipodes over a commutative ring).

[F3]

The unit map $η$ and the algebra maps $Δ, ε$ are unital, so $Δ (1_{A}) = 1_{A} \otimes 1_{A}$ and $η (ε (1_{A})) = 1_{A}$ (Bialgebras, counits and antipodes over a commutative ring).

[F4]

Each antipode $T$ satisfies $m (T \otimes id) Δ = η \circ ε = m (id \otimes T) Δ$ (Bialgebras, counits and antipodes over a commutative ring).

Proof

technique · Use uniqueness of a two-sided inverse in the convolution monoid of endomorphisms

1.1givenF1algebra

For $f, g \in {Hom}_{R} (A, A)$ define $f ⋆ g : = m \circ (f \otimes g) \circ Δ$ . Coassociativity of $Δ$ and associativity of $m$ give $(f ⋆ g) ⋆ h = f ⋆ (g ⋆ h)$ for all $f, g, h$ .

1.2F2algebra

The map $e : = η \circ ε$ is a two-sided unit for $⋆$ : for every $f$ and $a$ , $(e ⋆ f) (a) = \sum ε (a_{(1)}) f (a_{(2)}) = f (a)$ and $(f ⋆ e) (a) = \sum f (a_{(1)}) ε (a_{(2)}) = f (a)$ , by the two counit identities and $R$ -linearity of $f$ .

1.3givenF3F4algebra

Evaluating either antipode equation for $S$ at $1_{A}$ , and using $Δ (1_{A}) = 1_{A} \otimes 1_{A}$ and $e (1_{A}) = 1_{A}$ , gives $S (1_{A}) 1_{A} = e (1_{A}) = 1_{A}$ . Hence $S (1_{A}) = 1_{A}$ .

2.1step 1.1step 1.2F4algebra∎

By [F4], $S ⋆ id = e = id ⋆ S^{'}$ ; using [F1] and step 1.2, $S = S ⋆ e = S ⋆ (id ⋆ S^{'}) = (S ⋆ id) ⋆ S^{'} = e ⋆ S^{'} = S^{'}$ . Therefore the antipode is unique, and the stated bialgebra has at most one Hopf structure.

Depends on

Bialgebras, counits and antipodes over a commutative ring

Used by

The Drinfeld–Jimbo formulas define a Hopf algebra 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

Pavel Etingof and Mykola Semenyakin, A Brief Introduction to Quantum Groups (lecture notes, arXiv:2106.05252v3) (standard reference, not scraped)
Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin and Vera Serganova, Berkeley Lectures on Lie Groups and Quantum Groups (book-length lecture notes, last updated 18 January 2024) (standard reference, not scraped)