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The canonical algebra of a characteristic-zero fusion category is separable

Statement

Let $C$ be a fusion category over an algebraically closed field of characteristic zero. For the canonical Frobenius algebra $(K, μ, η, Δ, λ)$ constructed in The canonical algebra of a fusion category is Frobenius, set $t = λ μ Δ η$ . Then $t \neq 0$ and $s = t^{- 1} Δ$ is a $K$ -bimodule section of multiplication. In particular $K$ is separable, and $e = s η$ satisfies $μ e = η, (μ \otimes i d) (i d \otimes e) = (i d \otimes μ) (e \otimes i d) .$

Facts & Assumptions

[F1]

The canonical Frobenius algebra and its displayed bimodule identities are supplied by The canonical algebra of a fusion category is Frobenius.

[F2]

The squared-norm computation in Nonzero global dimension and the canonical Frobenius scalar identifies $λ μ Δ η$ with the nonzero global dimension of $C$ ; no pivotal structure is assumed.

Proof

Given: The Frobenius algebra and field hypotheses in the statement.

1.1F1givenalgebra

The bimodule equations imply that $u = μ Δ$ is a $K$ -bimodule endomorphism and $u = μ (i d \otimes u η)$ . Because ${Hom}_{D} (1, K) = k η$ , write $u η = t η$ . Thus $u = t {i d}_{K}$ ; applying $λ$ gives $t = λ μ Δ η$ since $λ η = 1$ .

2.1F1F2step 1.1algebra∎

Fact [F2] gives $t \neq 0$ . Therefore $s = t^{- 1} Δ$ is a bimodule map and $μ s = {i d}_{K}$ . Its value $e = s η$ obeys $μ e = η$ . Applying the left and right bimodule equations for $s$ to the unit identifies $(μ \otimes i d) (i d \otimes e)$ and $(i d \otimes μ) (e \otimes i d)$ with $s$ , proving centrality.

Depends on

Used by

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Sources

Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Sections 7.20–7.21 (standard reference, not scraped)