Centre of a Hopf algebra as $H$ -invariants

Let $H$ be a Hopf $K$ -algebra, regarded as a left $H$ -module under the adjoint action which we denote $𝑥 ▹ 𝑦$ . Then the space $H H$ of $H$ -invariants under this action is precisely the centre $Z (H)$ of $H$ .¹ #m/thm/hopf

Proof

If $𝑥 \in H H$ then for any $𝑦 \in H$ ,

𝑦 𝑥 = \sum (𝑦) (𝑦 (1) ▹ 𝑥) 𝑦 (2) = \sum (𝑦) 𝜖 (𝑦 (1)) 𝑥 𝑦 (2) = 𝑥 𝑦

by ^P1, so $𝑥 \in Z (H)$ . Conversely, if $𝑥 \in Z (H)$ then for any $𝑦 \in H$ ,

𝑦 ▹ 𝑥 = \sum (𝑦) 𝑦 (1) 𝑥 𝜎 (𝑦 (2)) = 𝑥 𝑦 (1) 𝜎 (𝑦 (2)) = 𝜖 (𝑦) 𝑥

so $𝑥 \in Z (H)$ .

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

1992. Casimir invariants for Hopf algebras, Lemma 1 ↩

Centre of a Hopf algebra as H-invariants

Footnotes

Centre of a Hopf algebra as $H$ -invariants