is generally not a coalgebra. However, the restricted dual A° of an algebra A is a coalgebra. It is defined as the space of linear forms on A vanishing on some finite-codimensional ideal (Sweedler 1969).
The coalgebra C is cocommutative if
Cocommutativity then expresses as:
In Sweedler’s notation coassociativity reads as:
We will sometimes write the iterated coproduct as:
Sometimes, we will even mix the two ways of using Sweedler’s notation for the iterated coproduct, in the case where we want to partially keep track of how we have constructed it (Dǎscǎlescu et al. 2001). For example,
With any vector space V, we can associate its tensor coalgebra Tc(V). It is isomorphic to T(V) as a vector space. The coproduct is given by the deconcatenation:
The counit is given by the natural projection of Tc(V) onto k.
Let C and D be the unital k-coalgebras. We put a counital coalgebra structure on C ⊗ D in the following way: the comultiplication is given by:
where
1.2.3. Convolution product
Let A be an algebra and C be a coalgebra (over the same field k). Then, there is an associative product on the space
In Sweedler’s notation, it reads:
The associativity is a direct consequence of both the associativity of A and coassociativity of C.
1.2.4. Bialgebras and Hopf algebras
A (unital and counital) bialgebra is a vector space ℋ endowed with a structure of unital algebra (m, ε) and a structure of counital coalgebra (Δ, ε), which are compatible. The compatibility requirement is that Δ is an algebra morphism (or equivalently that m is a coalgebra morphism), ε is an algebra morphism and u is a coalgebra morphism. It is expressed by the commutativity of the three following diagrams:
A Hopf algebra is a bialgebra ℋ together with a linear map S : ℋ → ℋ, called the antipode, such that the following diagram commutes:
In Sweedler’s notation, it reads:
In other words, the antipode is an inverse of the identity I for the convolution product on
A primitive element in a bialgebra ℋ is an element x, such that Δx = x⊗1 + 1⊗x. A grouplike element is a nonzero element x, such that Δx = x⊗x. Note that grouplike elements make sense in any coalgebra.
A bi-ideal in a bialgebra ℋ is a two-sided ideal, which is also a two-sided coideal. A Hopf ideal in a Hopf algebra ℋ is a bi-ideal J, such that S(J) ⊂ J.
1.2.5. Some simple examples of Hopf algebras
1.2.5.1. The Hopf algebra of a group
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.
1.2.5.2. Tensor algebras
There is a natural structure of cocommutative Hopf algebra on the tensor algebra T(V) of any vector space V. Namely, we define the coproduct Δ as the unique algebra morphism from T(V) into T(V) ⊗ T(V),