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Local Yang–Baxter operators on tensor powers

Definition

Let $C$ be a monoidal category, let $X \in C$ , let $R$ be a Yang–Baxter operator on $X$ (Yang–Baxter operators on an object), and let $n \geq 2$ . Write $X^{\otimes n}$ for the left-nested tensor power: $X^{\otimes 0} : = 1$ , $X^{\otimes 1} : = X$ and $X^{\otimes (k + 1)} : = X^{\otimes k} \otimes X$ for $k \geq 1$ ; write $1_{X}^{\otimes k}$ for the $k$ -fold tensor power of ${id}_{X}$ .

Work first in a strict model $C^{'}$ of $C$ , obtained from a monoidal equivalence as in Mac Lane strictification, and let $R^{'} ⁣ : X^{'} \otimes X^{'} \to X^{'} \otimes X^{'}$ be the transported Yang–Baxter operator. The local Yang–Baxter operators are the automorphisms of the $n$ -fold tensor power

$R_{i} : = 1_{X}^{\otimes (i - 1)} \otimes R \otimes 1_{X}^{\otimes (n - i - 1)} \in {Aut}_{C} (X^{\otimes n}), 1 \leq i \leq n - 1,$

where in the strict model the display is literal and each $R_{i}$ is the endomorphism of $X^{' \otimes n}$ acting as $R^{'}$ on the $i$ -th and $(i + 1)$ -st tensor factors and as the identity elsewhere.

Outside the strict model the same operators are obtained by conjugating the displayed word with canonical associativity isomorphisms: the composite

$X^{\otimes n} \overset{κ}{\to} X^{\otimes (i - 1)} \otimes (X \otimes X) \otimes X^{\otimes (n - i - 1)} \overset{1^{\otimes (i - 1)} \otimes R \otimes 1^{\otimes (n - i - 1)}}{\to} X^{\otimes (i - 1)} \otimes (X \otimes X) \otimes X^{\otimes (n - i - 1)} \overset{κ^{- 1}}{\to} X^{\otimes n},$

where $κ$ is any canonical morphism between the two parenthesised tensor words built from associators and unitors, is independent of the chosen $κ$ by Mac Lane coherence in canonical-map form (Mac Lane coherence in canonical-map form), and it is this composite that defines $R_{i}$ in $C$ .

Each $R_{i}$ is invertible, with inverse $1_{X}^{\otimes (i - 1)} \otimes R^{- 1} \otimes 1_{X}^{\otimes (n - i - 1)}$ (respectively its bracket-corrected conjugate): the displayed inverse is a two-sided inverse of the displayed word in the strict model, and conjugation by $κ$ preserves the inverse relation.

Depends on

Used by

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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)