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Conversely, using the dualities, we find for any pairing a morphism ψ ∈ Hom(A, ∨ A) such that
the operations are inverse.
A pairing is obviously non-degenerate, if and only if the morphism Φ is an isomorphism.
Similarly, invariance of the pairing amounts to the fact that Φ is a morphism of left modules.
This can be seen graphically and is relegated to an exercise.
✷
Definition 3.1.21
A Frobenius algebra in a rigid monoidal category C is an associative unital algebra A in C
together with the choice of one of the following three equivalent structures:
1. A (Δ, ɛ)-Frobenius structure on A.
2. A κ-Frobenius structure on A.
3. A Φ ρ -Frobenius structure on A.
Example 3.1.22.
It is instructive to write down explicitly a distinguished Frobenius algebra structure on the
group algebra K[G] of a finite group.
1. The bilinear form is defined on the distinguished basis by
κ(g, h) = δ gh,e for all g, h ∈ G .
This form is obviously non-degenerate and invariant, κ(gh, l) = δ ghl,e = κ(g, hl) for all
g, h, l ∈ G.
2. The corresponding Φ ρ -Frobenius structure is the morphism
Φ ρ : K[G] → K(G) = K[G] ∗
g ↦→ δ g −1
To show that this is indeed a morphism of left modules, we have to show Φ ρ (hg) = h ⇀
Φ ρ (g). Indeed, evaluating this on x ∈ G, we find
(h ⇀ δ g −1)(x) = δ g −1(xh) = δ g −1 h −1(x) = δ (hg) −1(x) for all x ∈ G .
3. We can finally deduce the (Δ F , ɛ F )-Frobenius structure, where we added an index F to
the Frobenius coproduct and counit to distinguish them from the Hopf coproduct and
counit. We find
ɛ F (g) = δ g,e and Δ F (x) = ∑ h∈G
gh −1 ⊗ h
which is indeed different from the coproduct and counit giving the Hopf algebra structure
on K[G]. Note that here, in contrast to the Hopf coproduct, the product in the group
enters.
We can now state:
Theorem 3.1.23.
Let H be a finite-dimensional Hopf-algebra with left integral λ ∈ H ∗ . Then H is a Frobenius
algebra with bilinear pairing
κ(h, h ′ ) := λ(h ∙ h ′ ) for h, h ′ ∈ H .
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