DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

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R-symbol as a channel exchange map

Definition

Let $a, b, c$ be simple objects in a unitary braided fusion category, with braiding $c_{a, b} : a \otimes b \to b \otimes a$ as in Braiding. The typed channel-exchange map on $V_{a b}^{c} = Hom (a \otimes b, c)$ is the pullback

$R_{a b}^{c} (f) : = f \circ c_{a, b}^{- 1} \in V_{b a}^{c} .$

Choose orthonormal bases in the two multiplicity spaces. The matrix of $R_{a b}^{c}$ in these bases is the R-symbol $R_{a b}^{c}$ . This pullback convention fixes the orientation of the coordinate exchange; reversing the braiding orientation replaces it by the inverse convention. If the spaces are zero-dimensional, the braiding isomorphism identifies both with $0$ and the R-symbol is the unique empty $0 \times 0$ matrix.

Depends on

Used by

Braid action on a fixed-charge fusion space Definition
Fibonacci fusion data and chiral convention Example
Ising fusion data and chiral convention Example
All-label unitary Ising normal data and ribbon coherence Lemma
The exact ell-four Ising gauge and parity chirality conversion Lemma
Hexagon in F/R coordinates Theorem
R-symbol unitarity Theorem

Dependency tree · two levels

9 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

David Penneys, Unitary Fusion Categories, Math 8800 lecture notes (standard reference, not scraped)
Chetan Nayak et al., Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)