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splits. Thus H = ker ɛ ⊕ I with I a left ideal of H.
Take z ∈ I and any h ∈ H. Then h−ɛ(h)1 ∈ ker ɛ. Since I is a left ideal, we have h∙z ∈ I.
The direct sum decomposition H = ker ɛ ⊕ I implies
Thus z ∈ I l (H) is a left integral in H.
I ∋ h ∙ z = (h − ɛ(h)1) ∙ z + ɛ(h)z = ɛ(h)z .
} {{ } } {{ }
∈ker ɛ
∈I
Since dim K I = 1, we may choose z ≠ 0. Then z /∈ ker ɛ and thus ɛ(I l (H)) ≠ 0.
2. Conversely, let I be a left integral and assume that ɛ(I) ≠ 0. Replacing I by a scalar
multiple, we can assume that ɛ(I) = 1. Then
s : K → H
λ ↦→ λI
obeys ɛ ◦ s(λ) = λɛ(I) = λ and is a morphism of left H modules, since I is a left integral,
so that the exact sequence (5) splits. Thus the trivial module is projective and the claim
follows from corollary 3.2.12.
✷
Example 3.2.14.
Consider the group algebra K[G] of a finite group G with two-sided integral I = ∑ g∈G
g. Then
ɛ(I) = ∑ g∈G
ɛ(g) = |G| .
Thus the group algebra K[G] is semisimple, if and only if char(K) ̸ | |G|.
Corollary 3.2.15.
If a finite-dimensional Hopf algebra is semisimple, it is unimodular.
Proof.
Since H is semisimple, we can choose a left integral t ∈ H such that ɛ(t) ≠ 0. Then for any
h ∈ H, we have
α(h)ɛ(t)t = α(h)t 2 = (th)t = t(ht) = ɛ(h)t 2 = ɛ(h)ɛ(t)t ,
where we used the definition of a left integral and of the distinguished group-like element α of
H ∗ . Since ɛ(t) ≠ 0, we have α(h) = ɛ(h) for all h ∈ H which implies unimodularity by corollary
3.1.16. ✷
We recall the notion of a separable algebra over a field K. To this end, let A be an associative
unital K-algebra. The algebra A e := A ⊗ A opp is called the enveloping algebra of A. If B is an
A-bimodule, it is a left module over A e by
(a 1 ⊗ a 2 ).b := (a 1 .b).a 2 .
Conversely, any A e -left module M carries a canonical structure of an A-bimodule with left
action a.m := (a ⊗ 1).m and right action m.a := (1 ⊗ a).m. Thus the categories of A e -left
modules and A-bimodules are canonically isomorphic as K-linear abelian categories.
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