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A twist on a braided rigid category is the same thing as a pivotal structure of Drinfeld type

Statement

Let $C$ be a braided tensor category and let $u_{X} : X \to X^{\lor \lor}$ be its Drinfeld isomorphism. For a natural automorphism $θ$ of the identity functor, put

$ψ_{X} : = u_{X} θ_{X} .$

Then $ψ$ is a pivotal structure if and only if $θ$ is a twist.

Facts & Assumptions

Given: A braided tensor category, its Drinfeld isomorphism $u$ , and a natural automorphism $θ$ of the identity functor.

[F1]

EGNO formula (8.35) and the sentence after it state exactly that $ψ_{X} = u_{X} θ_{X}$ is a tensor isomorphism $X \to X^{* *}$ if and only if $θ$ is a twist.

[L1]

The Drinfeld morphism exists, and in a braided tensor category it is an isomorphism (A braided rigid category has a Drinfeld morphism).

[L2]

A pivotal structure is precisely a monoidal natural isomorphism $id \Rightarrow (-)^{\lor \lor}$ (Pivotal structure).

[L3]

A twist is precisely a natural automorphism satisfying the double-braiding tensor law (Twist and ribbon structure).

Proof

technique · direct

1.1givenF1L1construct

By [L1], the family $u_{X}$ is invertible, so every natural isomorphism $ψ_{X} : X \to X^{\lor \lor}$ can be written uniquely as $u_{X} θ_{X}$ for a natural automorphism $θ$ of the identity.

2.1step 1.1L1L2L3

The tensor relation for $u$ from the Drinfeld-morphism theorem inserts exactly one double braiding between $u_{X \otimes Y}$ and $u_{X} \otimes u_{Y}$ . Therefore the condition that $ψ$ be monoidal is equivalent to the condition that $θ$ absorb that double braiding, namely $θ_{X \otimes Y} = (θ_{X} \otimes θ_{Y}) \circ c_{Y, X} \circ c_{X, Y},$ which is the twist law from [L3].

3.1step 2.1L2L3∎

Hence $ψ$ is a pivotal structure exactly when $θ$ is a twist.

Depends on

Used by

The explicit Fibonacci model has a positive spherical pivotal structure Lemma

Dependency tree · two levels

7 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, formula (8.35) and Proposition 8.10.6 (standard reference, not scraped)