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2. Let C be a coalgebra. A subspace I ⊂ C is a left coideal, if and only if C/I with the
natural map
Δ : C/I → C ⊗ C/I
inherited from the coproduct of C is a left comodule. There is an analogous statement
for right coideals.
3. Let S be a set and C := K[S] the coalgebra described above. A K-vector space M has
the structure of a C-comodule, if it is S-graded, i.e. if it can be written as a direct sum
of subspaces M s ⊂ M for s ∈ S:
M = ⊕ s∈S M s .
For an S-graded vector space M, set Δ M (m) := m ⊗ s for m ∈ M s . One directly checks
that this is a coaction. Conversely, given a C-comodule M, write Δ M (m) = ∑ s∈S m s ⊗ s.
The coassociativity of the action gives
(Δ M ⊗ id C ) ◦ Δ M (m) = ∑ s,t∈S(m s ) t ⊗ t ⊗ s
which has to be equal to
(id M ⊗ Δ) ◦ Δ M (m) = ∑ s∈S
m s ⊗ s ⊗ s
Thus (m s ) t = m s δ s,t , which implies Δ M (m s ) = m s ⊗ s. Denote by
M s := {m s | m ∈ M} .
Then the sum of the subspaces ⊕M s is direct: m ∈ M s ∩M t for s ≠ t implies m = m ′ s = m ′′
t
for some m ′ , m ′′ ∈ M. Then
Δ(m) = Δ(m ′ s) = m ′ s ⊗ s = m ⊗ s
and the same relation for t show that m = 0. Moreover, counitality of the coaction implies
m = id M (m) = (id M ⊗ ɛ) ◦ Δ M (m) = ∑ s∈S
m s ɛ(s) = ∑ s∈S
m s ,
so that M = ⊕ s∈S M s .
2.3 Bialgebras
Definition 2.3.1
1. A triple (A, μ, Δ) is called a bialgebra, if
• (A, μ) is an associative algebra, having a unit η : K → A.
• (A, Δ) is a coassociative coalgebra, having a counit ɛ : A → K.
• Δ : A → A ⊗ A is a map of unital algebras:
Δ(a ∙ b) = Δ(a) ∙ Δ(b)
for all a, b ∈ A
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