Sweedler's Hopf algebra


In mathematics, introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra H4 is a certain 4-dimensional quotient of it that is neither commutative nor cocommutative.

Definition

The following infinite dimensional Hopf algebra was introduced by. The Hopf algebra is generated as an algebra by three elements x, g, and g−1.
The coproduct Δ is given by
The antipode S is given by
The counit ε is given by
Sweedler's 4-dimensional Hopf algebra H4 is the quotient of this by the relations
so it has a basis 1, x, g, xg. Note that Montgomery describes a slight variant of this Hopf algebra using the opposite coproduct, i.e. the coproduct described above composed with the tensor flip on H4⊗H4.
Sweedler's 4-dimensional Hopf algebra is a quotient of the Pareigis Hopf algebra, which is in turn a quotient of the infinite dimensional Hopf algebra.