Hopf Algebra

Given a commutative ring , an R-algebra is a Hopf algebra if it has additional structure given by -algebra homomorphisms

(comultiplication) and

(counit) and an R-module homomorphism

(antipode) that satisfy the properties

1. Coassociativity:

2. Counitarity:

3. Antipode property:

where is the identity map on , is the multiplication in , and is the -algebra structure map for , also called the unit map.

Faà di Bruno's formula can be cast in a framework that is a special case of a Hopf algebra (Figueroa and Gracia-Bondía 2005).

Coassociativity means that the above diagram commutes, meaning if the arrows were reversed and were exchanged for , a diagram illustrating the associativity of the multiplication within would be obtained. And since embeds the ground ring into , the counit maps into . The counitarity property is similarly the dual of that satisfied by . With and , an algebra becomes a bialgebra, but it is the addition of the antipode that makes a Hopf algebra. The antipode should be thought of as an inverse on similar to that which exists within a group, and the antipode is an anti-homomorphism at the level of algebras and co-algebras, meaning that

where , which is called the switch map. Moreover, as with the inverse operation in a group, in many cases, the antipode is an involution.

The prototypical examples of Hopf algebras are group rings, where is a finite group and is a Hopf algebra via

for and extend by linearity to all of .

For general Hopf algebras, the comultiplication is given in Sweedler notation. That is, if then

which allows by coassociativity

to be unambiguously written.

Hopf algebras can be categorized into different types by dualizing the distinctions one makes between algebras. For example, if is commutative, this is equivalent to saying that satisfies the property that where is the switch map mentioned above. Likewise, a Hopf algebra is said to be cocommutative if , that is, if the above diagram commutes. Moreover, commutativity and cocommutativity are independent properties, and so Hopf algebras can be considered that satisfy one or the other, or both, or neither properties.

Additionally, just as the linear dual of an algebra is an algebra, the dual of a Hopf algebra is also a Hopf algebra, where the algebra structure of becomes the coalgebra structure of , and vice-versa, and the antipode for translates into an antipode for in a canonical fashion.