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R-symbol unitarity

Statement

In a unitary braided fusion category, every R-symbol matrix in orthonormal multiplicity bases is unitary.

Facts & Assumptions

Given: a unitary braided fusion category and simple charges $a, b, c$ .

[F1]

Unitary fusion category: the braiding is a unitary isomorphism, so $c_{a, b}^{†} = c_{a, b}^{- 1}$ .

[F2]

R-symbol as a channel exchange map: the channel map is $R_{a b}^{c} (f) = f \circ c_{a, b}^{- 1}$ from $V_{a b}^{c}$ to $V_{b a}^{c}$ .

[F3]

Fusion multiplicity spaces are finite-dimensional inner-product spaces: on either multiplicity space, $⟨ f, g ⟩ {id}_{c} = f \circ g^{†}$ .

Proof

1.1F1F2construct

The map $R_{a b}^{c}$ is invertible: its inverse sends $g \in V_{b a}^{c}$ to $g \circ c_{a, b}$ . Indeed, composing either way cancels $c_{a, b}^{- 1}$ with $c_{a, b}$ by [F2].

1.2F1F2F3algebra

For $f, g \in V_{a b}^{c}$ , the inner product of their images is $(f \circ c_{a, b}^{- 1}) \circ (g \circ c_{a, b}^{- 1})^{†} = f \circ c_{a, b}^{- 1} \circ c_{a, b} \circ g^{†} = f \circ g^{†}$ . The middle equality uses $(g \circ c^{- 1})^{†} = c \circ g^{†}$ from [F1] and contravariance of dagger. Thus [F3] shows that $R_{a b}^{c}$ preserves the inner product.

2.1F1F2step 1.1step 1.2∎

In the orthonormal bases used to define $R_{a b}^{c}$ , step 1.2 gives $(R_{a b}^{c})^{†} R_{a b}^{c} = I$ , and step 1.1 gives invertibility. Therefore $R_{a b}^{c}$ is unitary; when both spaces have dimension zero this is the empty identity matrix.

Depends on

Used by

Dependency tree · two levels

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Sources

David Penneys, Unitary Fusion Categories, Math 8800 lecture notes (standard reference, not scraped)