10.2 Hopf Algebras

In this Section, the relevant basics of Hopf algebras is summarized [98; 293].

Let be a complex unital algebra, i.e., a ?-ring. The tensor product of algebras is defined as that of vector spaces provided with the multiplication

Let us write the multiplication operation of the algebra as a ?-linear morphism

A coalgebra is defined as a vector space provided with the following linear morphisms:

• a coassociative comultiplication ,

• a counit ,

which obey the relations

(10.2.1)

As a shorthand, one writes

(10.2.2)

A comultiplication and a counit extend to tensor products pairwise, i.e.,

A bi-algebra ( , m, ?, ?) is defined as an associative algebra which is also a coalgebra so that

A Hopf algebra ( , m, ?, ?, S) is a bi-algebra endowed with a linear morphism , called the antipode, such that

It obeys the relations

where P : a ? b ? b ? a is the transposition operator. Let us note that, given a bi-algebra , there is a unique antipode, if any, such that becomes a Hopf algebra.

For the sake of brevity, we call ?, ? and S the co-operations of a Hopf algebra.

A Hopf algebra is said to be cocommutative if P ? ? = ?. If a Hopf algebra is commutative or cocommutative, then