Hopf ideal

Let $H$ be a Hopf algebra over $K$ . A Hopf ideal $𝐼 \leq H$ is a biïdeal which is additionally stable under the antipous, #m/def/ralg/hopf i.e. $𝜎 (𝐼) \subseteq 𝐼$ .

Given a Hopf ideal $𝐼$ one may construct a Quotient Hopf algebra $H / 𝐼$ .

Theorem

Let $I \leq H$ be an ideal such that

Δ I \subseteq I \otimes H + H \otimes I

and $I \cap K = 0$ . Then any of the following ensure $I$ is a Hopf ideal:¹

$H$ is module-finite.
$K$ is Noetherian, and $H / I$ is module-finite.
$K$ is Noetherian, and $H / I$ is a Commutative $K$ -monoid of finite type.

If instead $H$ is a Hopf algebra over $𝕂$ , then any of the following are sufficient.

$H / I$ is commutative.
$H$ is pointed.
$H$ is cocommutative.

Proof

#missing/proof See op. cit.

#state/develop | #lang/en | #SemBr

1978. Quotients of Hopf algebras ↩

Hopf ideal

Theorem

Footnotes