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Left-left Yetter–Drinfeld module

Definition

Let $H$ be a Hopf algebra over a field $k$ . A left-left Yetter–Drinfeld $H$ -module is a $k$ -vector space $V$ with a bilinear left action $H \otimes V \to V$ , $(h, v) \mapsto h \cdot v$ , satisfying $1_{H} \cdot v = v$ and $(h h^{'}) \cdot v = h \cdot (h^{'} \cdot v)$ , and a left coaction $δ : V \to H \otimes V$ , $δ (v) = \sum v_{- 1} \otimes v_{0}$ , satisfying

$\sum (Δ \otimes id) δ (v) = \sum (id \otimes δ) δ (v), (ε \otimes id) δ (v) = v,$

and the compatibility equation

$δ (h \cdot v) = \sum h_{1} v_{- 1} S (h_{3}) \otimes h_{2} \cdot v_{0} .$

A morphism is a $k$ -linear map preserving both action and coaction. For $f \in H^{*}$ define $f \cdot v : = \sum f (v_{- 1}) v_{0}$ . With convolution $(f * g) (x) = \sum f (x_{1}) g (x_{2})$ and opposite product $f ⊙ g = g * f$ , this is a left $H^{* o p}$ -action.

Facts & Assumptions

Given: A Hopf algebra $H$ with coproduct $Δ$ and counit $ε$ , and a vector space $V$ with a left $H$ -action and a counital, coassociative left $H$ -coaction. Prove that evaluation of the coaction defines a left $H^{* o p}$ -action.

[F1]

Bialgebras and bialgebra maps: $H$ is a coassociative counital coalgebra, with coproduct $Δ$ and counit $ε$ .

[F2]

Convolution of linear maps is associative and unital: convolution on linear maps from a coalgebra to an algebra is associative and unital.

[A1]

The displayed action is associative and unital, and the displayed coaction is coassociative and counital; these are the structure hypotheses in the definition.

Proof

1.1givenA1F1F2algebra

For each $v$ , the algebraic tensor $δ (v)$ is a finite sum, so evaluation defines $f \cdot v = \sum f (v_{- 1}) v_{0}$ . For $f, g \in H^{*}$ , coassociativity gives $f \cdot (g \cdot v) = \sum g (v_{- 1, 1}) f (v_{- 1, 2}) v_{0} = \sum (g * f) (v_{- 1}) v_{0} = (f ⊙ g) \cdot v$ . Thus evaluation reverses convolution exactly as required for the opposite product.

2.1givenA1F1step 1.1algebra∎

The dual algebra unit is $ε_{H}$ , and counitality gives $ε_{H} \cdot v = \sum ε_{H} (v_{- 1}) v_{0} = v$ . The action is therefore unital and associative for $H^{* o p}$ .

Depends on

Used by

Modules over the finite double are left-left Yetter–Drinfeld modules Theorem

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

G. A. García and C. Vay, Simple modules of small quantum groups at dihedral groups, Doc. Math. 29 (2024), §3.1.3 (standard reference, not scraped)
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik, Tensor Categories, §7.15 (standard reference, not scraped)