Sweedler's Hopf algebra
In mathematics, introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra "H" 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". The coproduct Δ is given by The antipode "S" is given by The counit ε is given by Sweedler's 4-dimensional Hopf algebra "H" 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 "H"⊗"H". 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.