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4. Let G be a finite group. Then K[G] is unimodular, with integrals
I l = I r = K ∑ g∈G
g .
Indeed, for I := ∑ h∈G h we have g.I = ∑ h∈G
gh = I = ɛ(g)I for all g ∈ G.
5. The dual KG of the group algebra K[G] is a commutative Hopf algebra. Suppose that
G is a finite group; then it can be identified with the commutative algebra of K-valued
functions on G. In this case, a right integral λ ∈ K[G] can be considered as an element
in the bidual, λ ∈ KG ∗ , i.e. a functional on functions of G. This is called a measure.
On the space of functions on a group G, we have a left action of G by translations:
L g :
KG → KG
with L g φ(h) = φ(hg). We compute, using that λ is a right integral:
λ(L g φ) = (L g )φ(λ) = φ(λ ∙ g) = φ(λ) = λ(φ) .
Thus the measure given by a right integral is left invariant.
6. The spaces of integrals for the Taft algebra are
and
The Taft algebra is thus not unimodular.
N−1
∑
I l = K g j x N−1
j=0
j=0
N−1
∑
I r = K ζ j g j x N−1 .
We need some actions and coactions of the Hopf algebra H on the dual vector space H ∗ .
Since we will use dualities, we assume H to be finite-dimensional.
Observation 3.1.10.
1. We consider H ∗ as a right H-comodule
ρ : H ∗ → H ∗ ⊗ H
f ↦→ f (0) ⊗ f (1)
with coaction derived from the coproduct in H:
〈p, f (1) 〉 ∙ 〈f (0) , h〉 = 〈p, h (1) 〉 ∙ 〈f, h (2) 〉 for all p ∈ H ∗ , h ∈ H .
Graphically, this definition is simpler to understand and the proof that ρ is a coaction is
then easy, see handwritten notes.
2. Consider for x ∈ H the K-linear endomorphism given by right multiplication with x
m x : H → H
h ↦→ h ∙ x
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