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and finally coassociativity reads
⎛
⎞
∑
⎝ ∑
(x (1) ) (1) ⊗ (x (1) ) (2)
⎠ ⊗ x (2) = ∑
(x) (x (1) )
(x)
⎛
⎞
x (1) ⊗ ⎝ ∑
(x (2) ) (1) ⊗ (x (2) ) (2)
⎠ .
(x (2) )
We abbreviate this element also by
∑
x(1) ⊗ x (2) ⊗ x (3) ≡ x (1) ⊗ x (2) ⊗ x (3) .
Lemma 2.2.4.
1. If C is a coalgebra, then the dual vector space C ∗ is an algebra, with multiplication from
m = Δ ∗ | C ∗ ⊗C ∗ and unit η = ɛ∗ .
Explicitly,
m(f ⊗ g)(c) = Δ ∗ (f ⊗ g)(c) = (f ⊗ g)Δ(c) for all f, g ∈ C ∗ and c ∈ C .
2. If the coalgebra C is cocommutative, then the algebra C ∗ is commutative.
Proof.
This is shown by dualizing diagrams, together with one additional observation: the dual of
Δ : C → C⊗C is a map (C⊗C) ∗ → C ∗ . Using the canonical injection C ∗ ⊗C ∗ ⊂ (C⊗C) ∗ → C ∗ ,
we can restrict Δ ∗ to the subspace C ∗ ⊗ C ∗ to get the multiplication on C ∗ .
✷
Remarks 2.2.5.
1. This shows a problem that occurs when we want to dualize algebras to obtain coalgebras:
since A ∗ ⊗ A ∗ is, in general, a proper subspace, A ∗ ⊗ A ∗ (A ⊗ A) ∗ , e.g. in the case of
infinite-dimensional algebras, the image of m ∗ : A ∗ → (A ⊗ A) ∗ may not be contained in
A ∗ ⊗ A ∗ . If A is finite-dimensional, this cannot happen, and A ∗ is a coalgebra.
2. For this reason, we denote by A o the finite dual of A:
A o := {f ∈ A ∗ | f(I) = 0 for some ideal I ⊂ A such that dim A/I < ∞} .
If A is an algebra, then A o can be shown to be a coalgebra, with coproduct Δ = m ∗ and
counit ɛ ∗ . If A is commutative, then A o is cocommutative.
We can dualize the notion of an ideal to get coalgebra structures on certain quotients:
Definition 2.2.6
Let C be a coalgebra.
1. A subspace I ⊂ C is a left coideal, if ΔI ⊂ C ⊗ I.
2. A subspace I ⊂ C is a right coideal, if ΔI ⊂ I ⊗ C.
3. A subspace I ⊂ C is a two-sided coideal, if
ΔI ⊂ I ⊗ C + C ⊗ I and ɛ(I) = 0 .
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