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Local braiding operators satisfy adjacent braid and distant commutation relations

Statement

Let $V$ be a left module over a quasitriangular Hopf algebra and let $n \geq 2$ . Set $\overset{ˇ}{R} : = c_{V, V}$ , with $c$ as in A quasitriangular structure gives and is recovered from a module braiding. For $1 \leq i < n$ , let $T_{i}$ be the operator on $V^{\otimes n}$ that acts by $\overset{ˇ}{R}$ on tensor positions $i, i + 1$ and by the identity on every other position, using the canonical associators of Associator naturality, pentagon, unit triangle and symmetry hexagons on elementary tensors. Then $T_{i} T_{i + 1} T_{i} = T_{i + 1} T_{i} T_{i + 1} (1 \leq i < n - 1), T_{i} T_{j} = T_{j} T_{i} (∣ i - j ∣ > 1) .$ These relations are the local operator facts used by later Hecke constructions; no quadratic relation is asserted.

Facts & Assumptions

Given: A field $k$ , a quasitriangular Hopf algebra $(H, R)$ , a left $H$ -module $V$ , and an integer $n \geq 2$ .

[F1]

A quasitriangular structure gives and is recovered from a module braiding: $\overset{ˇ}{R} = c_{V, V} = τ (ρ_{V} \otimes ρ_{V}) (R)$ is an invertible operator and satisfies the three-factor braid relation by the QYBE.

[F2]

Counit normalization and the Yang–Baxter identity for a quasitriangular structure: $R$ satisfies $R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$ .

[F3]

Associator naturality, pentagon, unit triangle and symmetry hexagons on elementary tensors: the canonical associators identify parenthesizations on pure tensors and preserve identities and compositions.

Proof

technique · direct

1.1givenF1F2F3algebra

On $V^{\otimes 3}$ write $T_{1} = τ_{12} R_{12}$ and $T_{2} = τ_{23} R_{23}$ . Conjugating a leg operator by an adjacent flip relabels its two tensor positions, so $T_{1} T_{2} T_{1} = τ_{12} τ_{23} τ_{12} R_{23} R_{13} R_{12}, T_{2} T_{1} T_{2} = τ_{23} τ_{12} τ_{23} R_{12} R_{13} R_{23} .$ The two flip products agree on every pure tensor by the permutation braid calculation, and [F2] identifies the R-products, proving the adjacent relation on three factors.

2.1step 1.1F3algebra

For any $n \geq 3$ and any adjacent positions $i, i + 1, i + 2$ , the corresponding three operators are the maps of step 1.1 tensored with identities before and after those positions. Functoriality of tensor maps and the canonical reassociations in [F3] preserve their equality, so $T_{i} T_{i + 1} T_{i} = T_{i + 1} T_{i} T_{i + 1}$ on $V^{\otimes n}$ .

3.1givenF1F3algebra∎

If $∣ i - j ∣ > 1$ , then $T_{i}$ and $T_{j}$ alter disjoint tensor positions. On each pure tensor, applying them in either order gives the same finite sum because each operator leaves every position used by the other unchanged; linearity extends equality to $V^{\otimes n}$ .

Depends on

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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)