Casimir invariant of a Hopf $𝕂$ -algebra

Let $H$ be a finite-dimensional Hopf $𝕂$ -algebra equipped with an invariant bilinear form $𝐵$ . Let $𝑅 \leq H a d H$ be an $H$ -submodule upon which $𝐵$ is nondegenerate, so that $𝜔 \in 𝑅 \otimes 𝑅$ is a coëvaluation. Then the Casimir invariant

Ω 𝑅, 𝐵 = 𝜇 (𝜔)

is in the centre of $H$ .¹ #m/thm/hopf In terms of string diagrams, $Ω 𝑅, 𝐵$ may be written as

Proof

We adopt the Einstein summation convention.

In light of Centre of a Hopf algebra as $H$ -invariants, it suffices to show $Ω : = Ω 𝑀, 𝐵$ satisfies

𝑥 ▹ Ω = 𝜖 (𝑥) Ω

for all $𝑥 \in H$ . Suppose $𝜔 = 𝑣 𝑖 \otimes 𝑣 𝑖$ . Applying ^P2 we have

𝑥 ▹ Ω = 𝑥 ▹ (𝑣 𝑖 𝑣 𝑖) = \sum (𝑥) (𝑥 (1) ▹ 𝑣 𝑖) (𝑥 (2) ▹ 𝑣 𝑖) = \sum (𝑥) ⟨ 𝑣 𝑗, 𝑥 (1) ▹ 𝑣 𝑖 ⟩ ⟨ 𝑥 (2) ▹ 𝑣 𝑖, 𝑣 𝑘 ⟩ 𝑣 𝑗 𝑣 𝑘 = \sum (𝑥) ⟨ 𝑣 𝑗, 𝑥 (1) ▹ 𝑣 𝑖 ⟩ ⟨ 𝑣 𝑖, 𝜎 (𝑥 (2)) ▹ 𝑣 𝑘 ⟩ 𝑣 𝑗 𝑣 𝑘 = \sum (𝑥) ⟨ 𝑣 𝑗, 𝑥 (1) ▹ 𝜎 (𝑥 (2)) ▹ 𝑣 𝑘 ⟩ 𝑣 𝑗 𝑣 𝑘 = \sum (𝑥) ⟨ 𝑣 𝑗, 𝑥 (1) 𝜎 (𝑥 (2)) ▹ 𝑣 𝑘 ⟩ 𝑣 𝑗 𝑣 𝑘 = 𝜖 (𝑥) ⟨ 𝑣 𝑗, 𝑣 𝑘 ⟩ 𝑣 𝑗 𝑣 𝑘 = 𝜖 (𝑥) Ω

as required.

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

1992. Casimir invariants for Hopf algebras ↩

Casimir invariant of a Hopf 𝕂-algebra

Footnotes

Casimir invariant of a Hopf $𝕂$ -algebra