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2. Note that, unlike in the case of bialgebras (which cannot be defined in any monoidal
category), neither the coproduct Δ nor the counit ɛ is an algebra morphism.
3. Concerning the Φ ρ -Frobenius structure, we remark that if Φ ρ ∈ Hom(A, ∨ A) is an isomorphism
between the left regular A-module (A, μ) and left A-module ∨ A, then dual
Φ ∨ ρ ∈ Hom(( ∨ A) ∨ , A ∨ ) = Hom(A, A ∨ )
on A is a left-module isomorphism Φ ρ ∈ Hom(A, ∨ A) between the left regular A-module
(A, μ) and right A-module ∨ A with the right dual action. This will be shown graphically.
It turns out that the three concepts are equivalent:
Proposition 3.1.19.
In a rigid monoidal category C the notions of a (Δ, ɛ)-Frobenius structure and of a κ-Frobenius
structure on an algebra (A, μ, η) are equivalent.
More concretely:
1. If (A, μ, η, Δ, ɛ) is an algebra with a (Δ, ɛ)-Frobenius structure, then (A, μ, η, κ ɛ ) with
is an algebra with κ-Frobenius structure.
κ ɛ := ɛ ◦ μ
2. If (A, m, η, κ) is an algebra with κ-Frobenius structure, then (A, μ, η, Δ κ , ɛ κ ) with
Δ κ := (id A ⊗ μ) ◦ (id A ⊗ Φ −1
κ ⊗ id A ) ◦ (b A ⊗ id A ) and ɛ κ := κ ◦ (id A ⊗ η)
with Φ κ ∈ Hom(A, A ∨ ) the morphism that exists by the assumption that κ is nondegenerate
is an algebra with (Δ, ɛ)-Frobenius structure.
Proof.
We present the proof that a (Δ, ɛ)-Frobenius structure gives a κ-Frobenius structure graphically.
The converse statement is relegated to an exercise.
✷
Proposition 3.1.20.
In a rigid monoidal category C the notions of a κ-Frobenius structure and of a Φ ρ -Frobenius
structure on an algebra (A, μ, η) are equivalent.
More specifically, for any algebra A in C the following holds:
1. There exists a non-degenerate pairing on A, if and only if A is isomorphic to ∨ A as an
object of C.
2. There exists an invariant pairing on A, if and only if there exists a morphism from A to
∨ A that is a morphism of left A-modules.
Proof.
Given a morphism Φ ∈ Hom C (A, ∨ A), we define a pairing on A by
κ Φ := ˜d A ◦ (id A ⊗ Φ) .
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