Читать книгу Algebra and Applications 2 - Группа авторов - Страница 14

Let G be a group, and let kG be the group algebra (over the field k). It is by definition the vector space freely generated by the elements of G: the product of G extends uniquely to a bilinear map from kG × kG into kG, hence, a multiplication m : kG ⊗ kG → kG, which is associative. The neutral element of G gives the unit for m. The space kG is also endowed with a counital coalgebra structure, given by:

and:

This defines the coalgebra of the set G: it does not take into account the extra group structure on G, as the algebra structure does.

PROPOSITION 1.3.– The vector space kG endowed with the algebra and coalgebra structures defined above is a Hopf algebra. The antipode is given by:

PROOF.– The compatibility of the product and the coproduct is an immediate consequence of the following computation: for any g, h ∈ G, we have:

Now, m(S ⊗ I)Δ(g) = g^-1 g = e and similarly for m(I ⊗ S)Δ(g). But, e = u ∘ ε(g) for any g ∈ G, so the map S is indeed the antipode. □

REMARK 1.1.– If G were only a semigroup, the same construction would lead to a bialgebra structure on kG: the Hopf algebra structure (i.e. the existence of an antipode) reflects the group structure (the existence of the inverse). We have S2 = I in this case; however, the involutivity of the antipode is not true for general Hopf algebras.