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A Yang–Baxter operator gives braid-group representations

Statement

Let $C$ be a monoidal category, let $X \in C$ , and let $R$ be a Yang–Baxter operator on $X$ . For every $n \geq 2$ there is a unique group homomorphism

$ρ_{n} ⁣ : B_{n} ⟶ {Aut}_{C} (X^{\otimes n}), ρ_{n} (σ_{i}) = R_{i},$

where $B_{n}$ is the braid group of The braid group by Artin presentation and $R_{1}, \dots, R_{n - 1}$ are the local operators of Local Yang–Baxter operators on tensor powers. The family $(ρ_{n})_{n \geq 2}$ is compatible with the standard inclusions $ι_{n} ⁣ : B_{n} \to B_{n + 1}$ , $ι_{n} (σ_{i}) = σ_{i}$ , in the sense that

$ρ_{n + 1} (ι_{n} (β)) = ρ_{n} (β) \otimes 1_{X} for all β \in B_{n} .$

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 $R_{1}, \dots, R_{n - 1}$ on $X^{\otimes n}$ .

[L1]

Each $R_{i}$ is an automorphism of $X^{\otimes n}$ , equal in a strict model to $1_{X}^{\otimes (i - 1)} \otimes R \otimes 1_{X}^{\otimes (n - i - 1)}$ and in general to its bracket-corrected conjugate; the correction is independent of the chosen canonical isomorphisms by coherence (Local Yang–Baxter operators on tensor powers).

[L2]

The local operators satisfy $R_{i} R_{j} = R_{j} R_{i}$ for $∣ i - j ∣ > 1$ and $R_{i} R_{i + 1} R_{i} = R_{i + 1} R_{i} R_{i + 1}$ for $1 \leq i \leq n - 2$ , with the bracket-corrected readings in the non-strict model (Local Yang–Baxter operators satisfy the Artin relations).

[L3]

For $n \geq 2$ the braid group $B_{n}$ is presented by generators $σ_{1}, \dots, σ_{n - 1}$ subject to the braid relations $σ_{i} σ_{i + 1} σ_{i} = σ_{i + 1} σ_{i} σ_{i + 1}$ and the distant-commutativity relations $σ_{i} σ_{j} = σ_{j} σ_{i}$ for $∣ i - j ∣ > 1$ (The braid group by Artin presentation).

[L4]

A map from the generators of a presented group to a group extends uniquely to a homomorphism if and only if the evaluation of every relator is the identity, and the extension is onto precisely when the images generate the target (Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group).

Proof

technique · direct

1.1L1given

The assignment lands in the automorphism group. By [L1] each $R_{i}$ is an automorphism of $X^{\otimes n}$ , so the assignment $σ_{i} \mapsto R_{i}$ is a map from the generating set ${σ_{1}, \dots, σ_{n - 1}}$ of $B_{n}$ into the group ${Aut}_{C} (X^{\otimes n})$ .

1.2L1L2L3algebra

The relators evaluate to the identity. By [L2] the values $R_{i}$ satisfy $σ_{i} σ_{i + 1} σ_{i} = σ_{i + 1} σ_{i} σ_{i + 1}$ and $σ_{i} σ_{j} = σ_{j} σ_{i}$ for $∣ i - j ∣ > 1$ with the bracket-corrected readings in the non-strict model. Since identities of morphisms in a strict model are preserved by the bracket correction of [L1] (the correction is by a common canonical isomorphism for the fixed tensor power), both families of defining relators of $B_{n}$ evaluate to the identity in ${Aut}_{C} (X^{\otimes n})$ .

2.1L3L4step 1.1step 1.2construct

Extension and uniqueness. By [L4] applied to the presentation [L3] and the map of step 1.1, whose relators evaluate to the identity by step 1.2, there is a unique homomorphism $ρ_{n} ⁣ : B_{n} \to {Aut}_{C} (X^{\otimes n})$ with $ρ_{n} (σ_{i}) = R_{i}$ . Uniqueness is the uniqueness clause of [L4]: the generators generate $B_{n}$ , so a homomorphism is determined by its values on them.

3.1L1L3L4step 2.1algebra

Compatibility with the standard inclusions. Fix $n \geq 2$ and consider the two maps $B_{n} \to {Aut}_{C} (X^{\otimes n + 1})$ given by $β \mapsto ρ_{n + 1} (ι_{n} (β))$ and by $β \mapsto ρ_{n} (β) \otimes 1_{X}$ . In the strict model the local operator at position $i$ for $n + 1$ strands is $1_{X}^{\otimes (i - 1)} \otimes R \otimes 1_{X}^{\otimes (n - i)} = (1_{X}^{\otimes (i - 1)} \otimes R \otimes 1_{X}^{\otimes (n - i - 1)}) \otimes 1_{X}$ , the local operator at position $i$ for $n$ strands tensored with $1_{X}$ ; in the non-strict model the same identity holds for the bracket-corrected operators by coherence, as in [L1]. Both displayed maps are homomorphisms and they agree on every generator $σ_{i}$ by this identity, so by the uniqueness clause of [L4] applied to the presentation [L3] they agree on all of $B_{n}$ .

4.1step 2.1step 3.1∎

Conclusion. Steps 2.1 and 3.1 give, for every $n \geq 2$ , the unique homomorphism $ρ_{n}$ with $ρ_{n} (σ_{i}) = R_{i}$ , compatible with the inclusions $ι_{n}$ . The construction uses only the Yang–Baxter relations and von Dyck's theorem; no choice principle is used.

Depends on

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17 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

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)