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in pictures
=
or in Sweedler notation
∑
(hg) (1) ⊗ (hg) (2) = ∑
h (1) g (1) ⊗ h (2) g (2) .
(hg)
(h)(g)
and Δ(1) = 1 ⊗ 1.
• ɛ : A → K is a map of unital algebras: ɛ(a ∙ b) = ɛ(a) ∙ ɛ(b). In pictures
=
and ɛ(1) = 1.
2. A K-linear map is said to be a bialgebra map, if it is both an algebra and a coalgebra
map.
Examples 2.3.2.
1. The tensor algebra T (V ) is a bialgebra with
Δ(v) = v ⊗ 1 + 1 ⊗ v and ɛ(1) = 1 and ɛ(v) = 0 for all v ∈ V .
We discuss its structure in more detail. Since ɛ is a morphism of algebras, one has
ɛ(v 1 ⊗ ∙ ∙ ∙ ⊗ v n ) = ɛ(v 1 ) ∙ . . . ∙ ɛ(v n ) = 0 .
This fixes the counit uniquely. Inductively, one shows
∑n−1
∑
Δ(v 1 . . . v n ) = 1⊗(v 1 . . . v n )+. . .+ (v σ(1) . . . v σ(p) )⊗(v σ(p+1) . . . v σ(n) )+. . .+(v 1 . . . v n )⊗1 .
p=1
σ
where the sum is over all (p, n − p)-shuffle permutations, i.e. all permutations σ ∈ S n for
which σ(1) < σ(2) < ∙ ∙ ∙ < σ(p) and σ(p + 1) < ∙ ∙ ∙ < σ(n).
Counitality is now a direct consequence of the explicit formulae for coproduct and counit.
Similarly, coassociativity can be derived. Alternatively, notice that the coproduct comes
from the diagonal map δ : v ↦→ v ⊗ v which obeys (δ ⊗ id V ) ◦ δ = (id V ⊗ δ) ◦ δ. Finally,
cocommutativity comes from the explicit formula for the coproduct, together with the
observation that (p, n − p)-shuffles are in bijection to (n − 1, p)-shuffles via a permutation
in S n that acts as (1, 2, . . . , n) ↦→ (p + 1, p + 2, . . . , n, 1, . . . p).
2. A direct calculation shows that the group algebra K[G] of a group G is a bialgebra. Note
that here we do not make use of the inverses in the group G, hence the statement even
holds for monoid algebras.
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