Trace form on a Hopf algebra of trace type

$H$ be a Hopf $𝕂$ -algebra, $𝑢 \in H$ be a tracelike, and $𝑉$ be a finite-dimensional $H$ -module affording the representation $𝜋 : H \to E n d 𝕂 𝑉$ . Then the $𝑢$ -trace form on $H$

⟨ 𝑥, 𝑦 ⟩ 𝑢, 𝑉 = t r 𝑉 (𝑢 𝑥 𝑦) = t r 𝜋 (𝑢 𝑥 𝑦)

is an $H$ -invariant bilinear form.¹ #m/thm/hopf

Proof

Using ^P2 we have

⟨ 𝑥 ▹ 𝑦, 𝑧 ⟩ = \sum (𝑥) t r 𝑉 (𝑢 𝑥 (1) 𝑦 𝑥 (2) 𝑧) = \sum (𝑥) t r 𝑉 (𝜎 2 (𝑥 (1)) 𝑢 𝑦 𝜎 (𝑥 (2)) 𝑧) = \sum (𝑥) t r 𝑉 (𝑢 𝑦 𝜎 (𝑥 (2)) 𝑧 𝜎 2 (𝑥 (1))) = ⟨ 𝑦, 𝜎 (𝑥) ▹ 𝑧 ⟩

as required.

Properties

$⟨ 𝑥, 𝑦 ⟩ = ⟨ 𝑦, 𝜎 2 (𝑥) ⟩$ .

Proof

Applying ^P2

⟨ 𝑥, 𝑦 ⟩ = t r 𝑉 (𝑢 𝑥 𝑦) = t r 𝑉 (𝜎 2 (𝑥) 𝑢 𝑦) = t r 𝑉 (𝑦 𝑢 𝜎 2 (𝑥)) = ⟨ 𝑦, 𝜎 2 (𝑥) ⟩

proving ^P1.

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

1992. Casimir invariants for Hopf algebras, p. 97 ↩

Trace form on a Hopf algebra of trace type

Properties

Footnotes