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2. Let (M, Δ M ) be a right H-comodule. The coinvariants of H on M are defined as the
K-vector space
M coH := {m ∈ M | Δ M (m) = m ⊗ 1} .
Examples 3.1.4.
1. If M is a right H-comodule, it can be considered as a left H ∗ -module. Then M H∗ = M coH .
2. Consider a group algebra, H = K[G]. For a left K[G]-module
For a K[G]-comodule the coinvariants
M K[G] = {m ∈ M | g.m = m for all g ∈ G} .
M coK[G] = M e
are the identity component of the G-graded vector space underlying the comodule.
3. For a module M over the universal enveloping algebra H = U(g),
M U(g) = {m ∈ M | x.m = 0 for all x ∈ g}.
The category of Hopf modules in itself is not of particular interest:
Theorem 3.1.5.
Let M be a right H-Hopf module. Then the multiplication map:
ρ : M coH ⊗ H → M
m ⊗ h ↦→ m.h
is an isomorphism of Hopf modules, where the left hand side has the structure of a trivial Hopf
module.
In particular, any Hopf module M is equivalent to a trivial Hopf module and thus a free
right H-module of rank dim K M coH .
Proof.
We perform the proof graphically, see separate file.
✷
Example 3.1.6.
Consider a Hopf module M over a group algebra K[G]. Since it is a comodule, M has the
structure of a G-graded vector space
M = ⊕ g∈G M g
with coaction Δ M (m g ) = m g ⊗ g for m g ∈ M g . Moreover, G acts on M. Since we have a Hopf
module, G acts such that Δ M (m.g) = Δ M (m).g. Thus for m g ∈ M g and h ∈ G, we have
Δ(m g .h) = m g .h ⊗ gh. Thus M g .h ⊂ M gh . Using the action of h −1 , we find the equality of
subspaces, M g .h = M gh . Thus the G-action permutes the homogeneous components and
M g = M 1 .g = M coK[G] .g .
This is exactly the statement of the fundamental theorem M ∼ = M coK[G] ⊗ K[G].
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