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S = S =
and in Sweedler notation
x (1) ∙ S(x (2) ) = ɛ(x) ∙ 1 = S(x (1) ) ∙ x (2) .
2. If an antipode exists, it is, as a two-sided inverse for an associative product, uniquely
determined:
S = S ∗ (ηɛ) = S ∗ (id H ∗ S ′ ) = (S ∗ id H ) ∗ S ′
= ηɛ ∗ S ′ = S ′ .
3. With H = (A, μ, η, Δ, ɛ, S) a finite-dimensional Hopf algebra, its dual H ∗ =
(A ∗ , Δ ∗ , ɛ ∗ , μ ∗ , η ∗ , S ∗ ) is a Hopf algebra as well.
4. We will see later that any morphism f : H → K of bialgebras between Hopf algebras
respects the antipode, f(S H h) = S K f(h) for all h ∈ H. It is thus a morphism of Hopf
algebras.
5. A subspace I ⊂ H of a Hopf algebra H is a Hopf ideal, if it is a biideal and if S(I) ⊂ I.
In this case, H/I with the structure induced from H is a Hopf algebra.
Examples 2.5.4.
1. If G is a group, the group algebra K[G] is a Hopf algebra with antipode
Indeed, we have for g ∈ G:
S(g) = g −1 for all g ∈ G .
μ ◦ (S ⊗ id) ◦ Δ(g) = g ∙ g −1 = ɛ(g)1 .
2. The universal enveloping algebra U(g) of a Lie algebra g is a Hopf algebra with antipode
Indeed, we have for x ∈ g:
S(x) = −x for all x ∈ g .
μ ◦ (S ⊗ id) ◦ Δ(x) = μ(−x ⊗ 1 + 1 ⊗ x) = −x + x = 0 = 1ɛ(x) .
In particular, the symmetric algebra over a vector space V is a Hopf algebra, since it is the
universal enveloping algebra of the abelian Lie algebra on the vector space V . Similarly,
the tensor algebra T V over a vector space V is a Hopf algebra, since it can be considered
as the enveloping algebra of the free Lie algebra on V .
If (A, μ A ) and (B, μ B ) are algebras, a map f : A → B is called an antialgebra map, if it
is a map of unital algebras f : A → B opp , i.e. if f(a ∙ a ′ ) = f(a ′ ) ∙ f(a) for all a, a ′ ∈ A and
f(1 A ) = 1 B .
Similarly, if (C, Δ C ) and (D, Δ D ) are coalgebras, a map g : C → D is called an
anticoalgebra map, if it is a counital coalgebra map g : C → C copp , i.e. if ɛ D ◦ g = ɛ C and
g(c) (2) ⊗ g(c) (1) = g(c (1) ) ⊗ g(c (2) ) .
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