Adjoint action

Let $H$ be a Hopf $K$ -algebra. Then $H$ can be regarded as an $H$ -module, ocasionally denoted $a d H$ , under the adjoint action #m/def/hopf

𝑥 ▹ 𝑦 = \sum (𝑥) 𝑥 (1) 𝑦 𝜎 (𝑥 (2)) .

In terms of string diagrams,

where the adjoint action is denoted in red.

Proof of

H

-module

Since $1 = 𝜎 (1)$ is grouplike, $1 ▹ 𝑥 = 𝑥$ for all $𝑥 \in H$ . For multiplicativity, note

as required.

Properties

$𝑥 𝑦 = \sum (𝑥) (𝑥 (1) ▹ 𝑦) 𝑥 (2)$ .
$𝜇 : H a d \otimes H a d \to H a d$ is an $H$ -morphism, i,e, $𝑥 ▹ 𝑦 𝑧 = \sum (𝑎) (𝑥 (1) ▹ 𝑦) (𝑥 (2) ▹ 𝑧)$

Proof

We have

proving ^P1.

We have

proving ^P2.

#state/tidy | #lang/en | #SemBr