Is it true that every commutative Hopf algebra is related to a Group in such a way that the co-multi

Aganippe76 2022-07-02 Answered
Is it true that every commutative Hopf algebra is related to a Group in such a way that the co-multiplication is originated from the multiplication of the group, the antipode from the inverse?
Making it more explicitly, can all commutative and co-comutative Hopf algebra H be written in this form: H = C [ G ], with the usual group algebra structure
η : 1 e , m : g h g h
the coalgebra structure
ε : g 1 , Δ : g g g
where all of the maps above are defined on the basis of group elements?
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Answers (1)

Zackery Harvey
Answered 2022-07-03 Author has 21 answers
Yes it is true. But not for the commutative hopf algebras in general. You need some more assumptions: first we need the field to be algebraically closed and of characteristic zero. You also need cocommutativity and finite dimensionality of the hopf algebra to have a full "correspondence".To be more precise:

it can be shown that there is an equivalence of Categories, between the Category of commutative, cocommutative, finite dimensional Hopf algebras H (over an algebraically closed field, of characteristic zero) and the category A b f i n of the finite, abelian groups. It is possible to construct fully faithful and essentially faithful functors between H and A b f i n .

Start from an object of H i.e. a commutative, cocommutative, finite dimensional Hopf algebra H, over an algebraically closed field, of characteristic zero. The set G ( H ) of its grouplike elements forms a finite abelian group i.e. an element of A b f i n . It is relatively easy to see that a hopf algebra morphism induces an abelian group homomorphism.So you get a functor
G : H A b f i n
On the other hand, start from a finite abelian group G and take its group hopf algebra k G. It is clearly commutative, cocommutative and finite dimensional, i.e. an object of H. On the other hand, an abelian group homomorphism induces -by linear extension, due to the universal property of the group algebra- a morphism of hopf algebras between the corresponding group hopf algebras.So you get a functor
F : A b f i n H
Now, it can be shown that:
G F = I d A b f i n F G I d H
Consequently, the functors G, F constitute an equivalence of the categories H, A b f i n .
In my understanding, it is actually this equivalence of categories, which inspired the introduction of the term quantum groups (implying that the hopf algebra theory may be considered as a kind of "quantum" generalization of the group theory).
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