DefinitionDefinition: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

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.

Leg notation and quasitriangular Hopf algebras

Facts & Assumptions

Given: A field $k$ and a Hopf algebra $H$ over $k$ .

[F1]

Hopf algebras and antipodes: $H$ is a unital associative $k$ -algebra with coproduct $Δ : H \to H \otimes H$ and counit $ε : H \to k$ satisfying the coalgebra identities.

[F2]

Scalars, tensor powers, the empty tensor, opposite algebras and finite sums: tensors are algebraic tensor products, every tensor is a finite sum of elementary tensors, and tensor powers have fixed left-associated parenthesization.

[F3]

The tensor product of $R$ -algebras has multiplication $(a \otimes b) (a^{'} \otimes b^{'}) = a a^{'} \otimes b b^{'}$ : multiplication on tensor-product algebras is componentwise, $(a \otimes b) (c \otimes d) = a c \otimes b d$ , with unit $1 \otimes 1$ .

[F4]

Associator naturality, pentagon, unit triangle and symmetry hexagons on elementary tensors: the canonical associators and flips are natural isomorphisms; parentheses may be changed through these specified maps.

Definition

Set $H^{\otimes 3} : = (H \otimes H) \otimes H$ . Define linear maps $ι_{12} (a \otimes b) = (a \otimes b) \otimes 1_{H}, ι_{13} (a \otimes b) = (a \otimes 1_{H}) \otimes b, ι_{23} (a \otimes b) = (1_{H} \otimes a) \otimes b .$ For $X \in H \otimes H$ , write $X_{i j} : = ι_{i j} (X)$ and $X_{21} : = τ_{H, H} (X)$ , where $τ_{H, H} (a \otimes b) = b \otimes a$ . Define $Δ^{o p} : = τ_{H, H} \circ Δ$ . In the second hexagon below, $α_{H, H, H}^{- 1}$ is the canonical reassociation $(H \otimes (H \otimes H)) \to ((H \otimes H) \otimes H)$ from [F4].

An algebraic quasitriangular structure on $H$ is an invertible element $R \in H \otimes H$ such that, for every $h \in H$ , $R Δ (h) = Δ^{o p} (h) R,$ and $(Δ \otimes {id}_{H}) (R) = R_{13} R_{23}, α_{H, H, H}^{- 1} ({id}_{H} \otimes Δ) (R) = R_{13} R_{12} .$ All products are in the componentwise tensor-product algebra, and $R$ is a finite algebraic tensor sum. The displayed formula with $α^{- 1}$ is the parenthesized form of $(id \otimes Δ) R = R_{13} R_{12}$ .

Verification

Given: A field $k$ , a Hopf algebra $H$ over $k$ , and the tensor-product algebra structures of [F3].

1.1givenF1F2construct

For each pair $i, j \in {12, 13, 23}$ the displayed assignment on elementary tensors is induced by a $k$ -bilinear map $H \times H \to H^{\otimes 3}$ ; the balanced tensor-product universal property therefore gives a well-defined linear map $ι_{i j} : H \otimes H \to H^{\otimes 3}$ . The flip $τ_{H, H}$ is induced by the balanced bilinear map $(a, b) \mapsto b \otimes a$ , so it is a well-defined linear map as well.

2.1givenF2F3algebrastep 1.1

On elementary tensors, componentwise multiplication gives $ι_{12} ((a \otimes b) (c \otimes d)) = (a c \otimes b d) \otimes 1_{H} = ι_{12} (a \otimes b) ι_{12} (c \otimes d),$ and the same calculation gives $(a c \otimes 1 \otimes b d)$ for $ι_{13}$ and $(1 \otimes a c \otimes b d)$ for $ι_{23}$ . Each map sends $1_{H} \otimes 1_{H}$ to $1_{H}^{\otimes 3}$ . Bilinearity and the spanning of elementary tensors extend these identities to all tensors, so each $ι_{i j}$ is a unital algebra homomorphism.

3.1givenF2F4algebra∎

The flip satisfies $τ_{H, H}^{2} (a \otimes b) = a \otimes b$ for every elementary tensor and hence $τ_{H, H}^{2} = {id}_{H \otimes H}$ . The right-parenthesized coproduct in the second hexagon is transported to the fixed left-associated $H^{\otimes 3}$ by the canonical associator [F4]. The flip is an involution of vector spaces, not asserted to be an algebra homomorphism. The formulas in the Definition therefore give well-defined leg tensors and opposite coproduct before any quasitriangular axioms are imposed.

Depends on

Used by

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Sources

Kalle Kytölä, Introduction to Hopf Algebras and Representations (Helsinki lecture notes, Spring 2011) (standard reference, not scraped)
Pavel Etingof and Mykola Semenyakin, A Brief Introduction to Quantum Groups, arXiv:2106.05252v3 (2024) (standard reference, not scraped)