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Local Yang–Baxter operators satisfy the Artin relations

Statement

Let $C$ be a monoidal category, let $X \in C$ , let $R$ be a Yang–Baxter operator on $X$ , let $n \geq 2$ , and let $R_{1}, \dots, R_{n - 1}$ be the local operators of Local Yang–Baxter operators on tensor powers, so that $R_{i} \in {Aut}_{C} (X^{\otimes n})$ is the local operator acting on the $i$ -th and $(i + 1)$ -st tensor factors. Then

$R_{i} R_{j} = R_{j} R_{i} whenever ∣ i - j ∣ > 1,$

$R_{i} R_{i + 1} R_{i} = R_{i + 1} R_{i} R_{i + 1} for 1 \leq i \leq n - 2.$

In the non-strict model the identities are those of the bracket-corrected local operators defined in Local Yang–Baxter operators on tensor powers.

Facts & Assumptions

Given: a monoidal category $C$ , an object $X$ , a Yang–Baxter operator $R$ on $X$ , an integer $n \geq 2$ , and the local operators $R_{1}, \dots, R_{n - 1}$ .

[L1]

In a strict model the local operator is $R_{i} = 1_{X}^{\otimes (i - 1)} \otimes R \otimes 1_{X}^{\otimes (n - i - 1)}$ , and in general it is the bracket-corrected conjugate of that word; every $R_{i}$ is invertible (Local Yang–Baxter operators on tensor powers).

[L2]

The Yang–Baxter operator satisfies the cubic equation $(R \otimes 1_{X}) (1_{X} \otimes R) (R \otimes 1_{X}) = (1_{X} \otimes R) (R \otimes 1_{X}) (1_{X} \otimes R)$ (Yang–Baxter operators on an object).

Proof

technique · direct

1.1L1givenalgebra

Far commutativity in the strict model. Suppose first that $C$ is strict and $∣ i - j ∣ > 1$ . The endomorphisms $R_{i}$ and $R_{j}$ are tensor products of the identity with the single factor $R$ inserted at positions ${i, i + 1}$ respectively ${j, j + 1}$ ; these supports are disjoint because $∣ i - j ∣ > 1$ . Tensoring the two words and using functoriality of the tensor product, both $R_{i} R_{j}$ and $R_{j} R_{i}$ are the same tensor product of identities with two copies of $R$ at positions ${i, i + 1}$ and ${j, j + 1}$ (in the two possible orders of composition); hence $R_{i} R_{j} = R_{j} R_{i}$ .

1.2L1L2givenalgebra

The adjacent relation in the strict model. Suppose $C$ is strict and $1 \leq i \leq n - 2$ . Every tensor factor outside positions $i, i + 1, i + 2$ carries only identities in each of the words $R_{i} R_{i + 1} R_{i}$ and $R_{i + 1} R_{i} R_{i + 1}$ , so both sides are $1_{X}^{\otimes (i - 1)}$ tensored with an endomorphism of the three middle factors tensored with $1_{X}^{\otimes (n - i - 2)}$ . On those three middle factors the two sides are $(R \otimes 1_{X}) (1_{X} \otimes R) (R \otimes 1_{X})$ and $(1_{X} \otimes R) (R \otimes 1_{X}) (1_{X} \otimes R)$ , which are equal by the cubic equation [L2]. Since tensoring equal morphisms with identities gives equal morphisms, $R_{i} R_{i + 1} R_{i} = R_{i + 1} R_{i} R_{i + 1}$ .

2.1L1step 1.1step 1.2algebra

The non-strict model. Let $E : C \to C^{'}$ be the strong monoidal equivalence used in [L1], put $X^{'} = E (X)$ , and let $J_{n} : X^{' \otimes n} \to E (X^{\otimes n})$ be its iterated tensor constraint. Transport $R$ as $R^{'} = J_{2}^{- 1} E (R) J_{2}$ . Naturality and associativity coherence of the constraints give $E (R_{i}) = J_{n} R_{i}^{s t r} J_{n}^{- 1}$ : apply $E$ to the bracket-corrected composite of [L1] and use the strong monoidal constraint at each tensor product. Thus $E$ sends each proposed Artin identity to the corresponding identity of steps 1.1 and 1.2, conjugated by the single isomorphism $J_{n}$ . Since an equivalence is faithful, the two identities hold in $C$ . Every comparison here is a morphism in $C^{'}$ .

3.1step 1.1step 1.2step 2.1∎

Conclusion. Steps 1.1 and 1.2 prove the two families of identities in the strict model, and step 2.1 transports them to the bracket-corrected operators of the non-strict model. This proves the lemma. The argument is a finite computation in the tensor product and uses no choice principle.

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Sources

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (AMS Mathematical Surveys and Monographs 205), author's final version (standard reference, not scraped)
V. G. Turaev, Quantum Invariants of Knots and 3-Manifolds (de Gruyter Studies in Mathematics 18, 1994) (standard reference, not scraped)