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It follows that the components Λ i (1)
of any representation of Δ(Λ)
Δ(Λ) = ∑ i
Λ i (1) ⊗ Λ i (2) ∈ H ⊗ H
form a generating system of H. We can thus find a form for Δ(Λ) such that the components
(Λ i (1) ) form a basis. Thus (S(Λ (1)), Λ (2) ) form a dual basis for the standard Frobenius
structure on on the Hopf algebra H given by λ.
2. Assume now that H is semisimple. By Maschke’s theorem κ := ɛ(Λ) ≠ 0. Then
is a separability idempotent. Indeed,
e := κ −1 ∙ S(Λ (1) ) ⊗ Λ (2) ∈ H ⊗ H
μ(e) := κ −1 S(Λ (1) ) ∙ Λ (2) = κ −1 1 H ɛ(Λ) = 1 H
by the defining property of the antipode. The Casimir property of a separability idempotent
follows directly from observation 3.2.20.4, since it is built from a pair of dual
bases.
3.3 Powers of the antipode
Observation 3.3.1.
Let V be a finite-dimensional K-vector space. Under the canonical identification
V ∗ ⊗ V → End K (V )
β ⊗ v ↦→ (w ↦→ β(w)v)
the trace becomes Tr(β ⊗ v) = β(v). Indeed, with dual bases (e i ) i∈I of V and (e i ) i∈I of V ∗ , we
find β = ∑ i β ie i and v = ∑ i vi e i . The corresponding linear map is ∑ i,j β iv j e j ⊗ e i which has
trace ∑ i β iv i = β(v).
Lemma 3.3.2.
Let H be a finite-dimensional Hopf algebra with λ ∈ I l (H ∗ ) and a right integral Λ ∈ H such
that λ(Λ) = 1. Let F be a linear endomorphism of H. Then
Tr(F ) = 〈λ, F (Λ (2) )S(Λ (1) )〉 .
Proof.
We know by remark 3.2.21.1 that for all x ∈ H, we have
F (x) = 〈λ, F (x)S(Λ (1) )〉Λ (2) .
Thus under the identification H ∗ ⊗ H ∼ = End(H), the endomorphism F corresponds to
〈λ, F (−)S(Λ (1) )〉 ⊗ Λ (2) ;
thus
Tr(F ) = 〈λ, F (Λ (2) )S(Λ (1) )〉 .
✷
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