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3. A coalgebra map is a linear map
such that the equations
ϕ : C → C ′ ,
Δ ′ ◦ ϕ = (ϕ ⊗ ϕ) ◦ Δ and ɛ ′ ◦ ϕ = ɛ
hold. Pictorially,
ϕ
=
ϕ
ϕ
ϕ
= = ɛ
Examples 2.2.2.
1. Let S be any set and C = K[S] the free K-vector space with basis S. Then C becomes a
coassociative counital coalgebra with coproduct given by the diagonal map Δ(s) = s ⊗ s
and ɛ(s) = 1 for all s ∈ S. It is cocommutative.
2. In particular, the group algebra K[G] for any group G with the maps Δ, ɛ discussed above
can be easily shown to be a coalgebra which is cocommutative.
3. The universal enveloping algebra U(g) of any Lie algebra with the maps Δ, ɛ discussed
above will be shown to be a coalgebra which is cocommutative. (This is easier to do, once
we have stated compatibility conditions between product and coproduct.)
Remarks 2.2.3.
1. The counit is uniquely determined, if it exists.
2. The following notation is due to Heyneman and Sweedler and frequently called Sweedler
notation: let (C, Δ, ɛ) be a coalgebra. For any x ∈ C, we can find finitely many elements
x ′ i ∈ C and x ′′
i ∈ C such that
Δ(x) = ∑ x ′ i ⊗ x ′′
i .
i
We write this by dropping summation indices as
Δ(x) = ∑ (x)
x (1) ⊗ x (2)
and sometimes even omit the sum. In this notation, counitality reads
∑
ɛ(x (1) )x (2) = ∑ x (1) ɛ(x (2) ) = x for all x ∈ C,
(x)
(x)
and cocommutativity
∑
x (1) ⊗ x (2) = ∑
(x)
(x)
x (2) ⊗ x (1)
for all x ∈ C
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