In mathematics, introduced an example of an infinite-dimensional Hopf algebra

Hopf algebra

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...

, 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 h with the relations gh = 1 (so h is denoted by g⁻¹).

The coproduct Δ is given by

Δ(g) = g⊗g, Δ(x) = 1⊗x + x⊗g

The antipode S is given by

S(x) = –g^–1x, S(g) = g

The counit ε is given by

ε(x)=0, ε(g) = 1

Sweedler's 4-dimensional Hopf algebra H₄ is the quotient of this by the relations

x² = 0, g² = 1, gx = –xg

so it has a basis 1, x, g, xg .