In mathematics, Moss E. Sweedler (1969, p. 89–90) 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.
The following infinite dimensional Hopf algebra was introduced by Sweedler (1969, pages 89–90).
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 (Montgomery 1993, p.8).
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.