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is an exact sequence in C, then
is exact in C as well.
0 → U ⊗ X → V ⊗ X → W ⊗ X → 0
Proof.
This follows from lemma 3.2.9, since the functor of tensoring with a rigid object has a left and
a right adjoint by example 2.5.22.
✷
For the following propositions, the reader might wish to keep the category C = H−mod fd
of finite-dimensional modules over a finite-dimensional Hopf algebra in mind.
Lemma 3.2.11.
Let C be a rigid abelian tensor category. Let P be a projective object and let M be any object.
Then the object P ⊗ M is projective.
Proof.
By rigidity, we have adjunction isomorphisms
Hom(P ⊗ M, N) ∼ = Hom(P, N ⊗ M ∨ ) .
Thus the functor Hom(P ⊗ M, −) is isomorphic to the concatenation of the functor − ⊗ M ∨
(which is exact by lemma 3.2.10) with the functor Hom(P, −) which is exact by property 4 of
the projective object P .
✷
Corollary 3.2.12.
A K-Hopf algebra is semi-simple, if and only if the trivial module (K, ɛ) is projective.
Proof.
If the trivial module I = (K, ɛ) – which is the tensor unit in H−mod – is projective, then by
lemma 3.2.11 any module M ∼ = M ⊗ I is projective. The converse is trivial.
✷
Theorem 3.2.13 (Maschke).
Let H be a finite-dimensional Hopf algebra. Then the following statements are equivalent:
1. H is semisimple.
2. The counit takes non-zero values on the space of left integrals, ɛ(I l (H)) ≠ 0.
Proof.
1. Suppose that H is semisimple. Then any module is projective, in particular the trivial
module. Thus the exact sequence of left H-modules
(0) → ker ɛ → H ɛ → K → (0) (5)
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