$H$ -invariant

Let $H$ be a $K$ -bimonoid and $𝑉$ be an $H$ -module. An $H$ -invariant is an element $𝑣 \in 𝑉$ for which

𝑥 𝑣 = 𝜖 (𝑥) 𝑣

for all $𝑥 \in H$ , #m/def/hopf i.e. $𝑣$ transforms under the trivial action. The set of invariants

𝑉 H : = {𝑣 \in 𝑉 : ∣ : \forall 𝑥 \in H . 𝑥 𝑣 = 𝜖 (𝑥) 𝑣}

is then a trivial $H$ -submodule.

Proof

Since $𝑉 H$ is clearly a $K$ -submodule, and the action of $H$ factors through the action of $K$ .

Special cases

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