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The finite dual is a right regular Hopf module

Statement

Let $H$ be a finite-dimensional Hopf algebra over a field $k$ . For $f \in H^{*}$ and $h, x \in H$ , define

(f ↼ h) (x) = f (x S (h)) .

For $f \in H^{*}$ , define $ρ (f) = \sum f_{0} \otimes f_{1} \in H^{*} \otimes H$ by the requirement

\sum p (f_{1}) f_{0} (x) = \sum p (x_{1}) f (x_{2}) (p \in H^{*}, x \in H) .

This requirement determines a unique tensor. With this right action and right coaction, $H^{*}$ is a right-right Hopf module. Its coinvariants are exactly the left integral functionals: $(H^{*})^{co H} = \int_{H^{*}}^{ℓ}$ .

Facts & Assumptions

Given: A finite-dimensional Hopf algebra $H$ over $k$ ; its dual Hopf algebra $H^{*}$ ; $f \in H^{*}$ ; and $h, x \in H$ , $p, q \in H^{*}$ .

[F1]

Left and right integrals and integral functionals: left integral functionals satisfy $p * f = p (1) f$ for every $p \in H^{*}$ , with convolution dual to $Δ_{H}$ .

[F2]

Right-right Hopf modules and coinvariants: a right-right Hopf module has a right $H$ -action and right $H$ -coaction satisfying $ρ (m h) = \sum m_{0} h_{1} \otimes m_{1} h_{2}$ ; its coinvariants satisfy $ρ (m) = m \otimes 1$ .

[F3]

Finite tensor duality and basis-independent coevaluation: for finite-dimensional spaces, tensor duality identifies bilinear pairings with unique tensors, and the finite dual basis separates tensors.

[F4]

Uniqueness and anti-multiplicativity and anti-comultiplicativity of the antipode: the antipode satisfies $S (1) = 1$ , $S (h g) = S (g) S (h)$ , and $Δ (S (h)) = \sum S (h_{2}) \otimes S (h_{1})$ . The antipode equations in Hopf algebras and antipodes give $\sum S (h_{1}) h_{2} = \sum h_{1} S (h_{2}) = ε (h) 1$ .

[F5]

The dual of a finite-dimensional Hopf algebra is a Hopf algebra: since $H$ is finite-dimensional, its full linear dual has the transposed Hopf structure, with convolution product and counit $f \mapsto f (1)$ .

Proof

1.1givenF3algebra

Fix a basis $(e_{i})$ of $H$ with dual basis $(e_{i}^{*})$ . The tensor $ρ (f) = \sum_{i} e_{i}^{*} \otimes (\sum (e_{i})_{1} f ((e_{i})_{2}))$ has the required pairing, since $x = \sum_{i} e_{i}^{*} (x) e_{i}$ . Uniqueness follows from finite tensor separation [F3]. The assignment $f \mapsto ρ (f)$ is linear because its defining pairing is linear in $f$ and the representing tensor is unique.

1.2givenF2F3F4algebra

The formula defines a right action: $((f ↼ h) ↼ g) (x) = f (x S (g) S (h)) = f (x S (h g)) = (f ↼ (h g)) (x)$ by anti-multiplicativity [F4], and $f ↼ 1 = f$ . Pair each side of $ρ (f ↼ h) = \sum (f_{0} ↼ h_{1}) \otimes f_{1} h_{2}$ with $x, p$ . The left side gives $\sum p (x_{1}) f (x_{2} S (h))$ . The right side gives $\sum p_{1} (x_{1} S (h_{2})) p_{2} (h_{3}) f (x_{2} S (h_{1})) = \sum p (x_{1} S (h_{2}) h_{3}) f (x_{2} S (h_{1})) = \sum p (x_{1}) f (x_{2} S (h))$ , because $Δ p = \sum p_{1} \otimes p_{2}$ is dual to multiplication, $Δ (x S (h_{1})) = \sum x_{1} S (h_{2}) \otimes x_{2} S (h_{1})$ by anti-comultiplicativity [F4], and $\sum S (h_{2}) h_{3} = ε (h_{2}) 1$ . The pairings agree; finite tensor separation [F3] proves compatibility.

2.1step 1.1F3F5algebra

The counit identity follows by evaluating $ρ (f)$ against $p = ε$ : $\sum ε (f_{1}) f_{0} (x) = \sum ε (x_{1}) f (x_{2}) = f (x)$ . For coassociativity, pair both $(ρ \otimes i d) ρ (f)$ and $(i d \otimes Δ) ρ (f)$ with $x, p, q$ . The first pairing is $(p * (q * f)) (x)$ by the defining equation twice; the second is $((p * q) * f) (x)$ . Convolution is associative because $Δ$ is coassociative, so the pairings agree for all $x, p, q$ , and finite tensor separation [F3] gives equality of the tensors.

3.1step 1.1F1F5algebra∎

The coaction coinvariant condition is equivalent, after pairing with every $p, x$ , to $(p * f) (x) = p (1) f (x)$ . Since $ε_{H^{*}} (p) = p (1)$ , this is exactly the definition of a left integral functional in [F1]; hence $(H^{*})^{co H} = \int_{H^{*}}^{ℓ}$ .

Depends on

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Sources

Paolo Saracco, Notes on Hopf Modules, Integrals and Frobenius Hopf Algebras (course notes, 2019) (standard reference, not scraped)
Erasmus Mini-Course on Hopf Algebras (lecture notes) (standard reference, not scraped)